https://wiki.haskell.org/api.php?action=feedcontributions&user=Blackout&feedformat=atomHaskellWiki - User contributions [en]2020-07-11T03:13:43ZUser contributionsMediaWiki 1.27.4https://wiki.haskell.org/index.php?title=ZuriHac2016&diff=60927ZuriHac20162016-07-20T18:09:25Z<p>Blackout: /* David Luposchainsky */ Updated (outdated) talk abstract</p>
<hr />
<div>[[Image:ZuriHac2016S.png|right]]<br />
<br />
----<br />
{|<br />
|When:<br />
|Friday 22th of July 2016 - Sunday 24th of July 2016<br />
|-<br />
|Where:<br />
|Google, Zurich, Switzerland<br />
|}<br />
----<br />
<br />
'''Important:'''<br><br />
Switzerland has [http://en.wikipedia.org/wiki/AC_power_plugs_and_sockets#Swiss_SEV_1011_.28Type_J.29 its own power sockets]. We can't provide converters for everybody so make sure to bring one along. Do note that the [http://en.wikipedia.org/wiki/Europlug Europlug] will fit in a Swiss power socket.<br />
<br />
&nbsp;<br />
<br />
== About ==<br />
<br />
End of July 2016, the [http://www.meetup.com/haskellerz Zurich Haskell Meetup group] will organize ZuriHac 2016, a three day Haskell Hackathon hosted at the Google offices. This is the [https://wiki.haskell.org/ZuriHac fifth Haskell Hackathon in Zurich].<br />
<br />
The Haskell Hackathon is an international, grassroots collaborative coding festival with a simple focus: build and improve Haskell libraries, tools, and infrastructure.<br />
<br />
This is a great opportunity to meet your fellow Haskellers in real life,<br />
find new contributors for your project, improve existing libraries and tools or<br />
even start new ones!<br />
<br />
This event is open to any experience level, from beginners to gurus. In fact, one of the goals is to bring beginners in contact with experts so that the former can get a quick start in the Haskell community. We will have a dedicated beginners' track, and there are going to be mentors on site whom you can directly approach during the whole event with any Haskell-related question that might pop up.<br />
<br />
== Sponsors ==<br />
<br />
[[File:Google.png|230px]]<br />
<br />
[http://www.google.com/about/jobs/locations/zurich/ Google Switzerland] hosts the Hackathon.<br />
<br />
== Venue notice ==<br />
<br />
Visitors are not allowed to take any photos in the venue. No distribution of any Google work-related information visible inside the building. Visitors should always have a visitor sticker visible. Guests can not be left unattended.<br />
<br />
== News ==<br />
<br />
;30th of April 16 at 12pm (UTC+2)<br />
:Registration is going to be opened<br />
<br />
;28th of April 16<br />
:Added wiki page<br />
<br />
== Registration ==<br />
<br />
Unfortunately we have reached our maximum capacity. If you want to be queued into the wait list please still [https://docs.google.com/forms/d/1Dh2L8h9Ub_3Md-53ViwCNnbpSHn6B0nd9zTX3H6w334/viewform?c=0&w=1&usp=mail_form_link fill out this form]. In case that somebody cancels we will fill up the seats from the wait list in first come first serve order and let you know. Do not book anything before we have confirmed a seat for you (in a second confirmation email) though.<br />
<br />
== Attendees ==<br />
Here is the [[/Attendees|list of participants]] who have signed up for ZuriHac 2016 and have chosen their name to be listed there.<br />
<br />
== Location ==<br />
[[File:ZurichMontage.jpg|300px|right]]<br />
<br />
'''Zurich, Switzerland'''<br />
<br />
Here is a map with [https://www.google.com/maps/d/edit?mid=11179LcsFVKk-gIjLv57dVxI1yfg map with all important locations].<br />
<br />
To learn more about the city, please see the [http://en.wikipedia.org/wiki/Zurich '''Wikipedia'''] and [http://wikitravel.org/en/Zurich '''Wikitravel'''] articles.<br />
<br />
See here for [http://www.zvv.ch/ public transport in the Zurich area].<br />
<br />
'''A note on day tickets:'''<br />
<br />
Day tickets in Zurich run for 24 hours and can be used on all forms of transport in zone 110 that is bus, train, tram or even ship.<br />
<br />
If you arrive via airplane, please note that the airport is outside of the zone 110, which only covers the city of Zurich itself. So you need to upgrade your ticket to include for the time of your travel into the city (around 15 min) one additional zone. <br />
<br />
You can buy tickets from [http://www.zvv.ch/en/tickets/automatic-ticket-machine/index.html the ticket machines].<br />
<br />
== Accommodation ==<br />
<br />
We recommend<br />
[https://www.google.com/maps/d/edit?mid=11179LcsFVKk-gIjLv57dVxI1yfg the following accommodation options]. Please note that although accommodations around [https://goo.gl/maps/G1nsFye8Ndp Langstrasse] tend to be cheap that area is usually very loud at night especially during weekends.<br />
<br />
== Schedule ==<br />
{|<br />
|-<br />
! !! Friday !! Saturday !! Sunday<br />
|-<br />
| 9am || doors open || doors open || doors open<br />
|-<br />
| 10am || '''Welcome & Project Introduction''' || ''Edward Kmett'', '''tba''' (Key Note) || ''Andres Löh'', '''Generic (and type-level) programming with Generics-Sop''' (Key Note)<br />
|-<br />
| 10:30am || || || ''Andrey Mokhov'', '''Meet Hadrian: a new build system for GHC''' (Talk)<br />
|-<br />
| 11am || ''Bas van Dijk'', '''FP at LumiGuide''' (Key Note) || || <br />
|-<br />
| 1pm || || lunch || lunch<br />
|-<br />
| 2pm || lunch || ||<br />
|- <br />
| 4pm || || || '''Project Presentation & Closing'''<br />
|-<br />
| 5pm || || ''David Luposchainsky'', '''Low-level Haskell: an interactive tour through the STG''' (Talk) ||<br />
|-<br />
| 5:30pm || ''Francesco Mazzoli'', '''Parallelizing and distributing scientific software in Haskell''' (Talk) || ||<br />
|-<br />
| 6pm || ''Luka Rahne'', '''CλaSH - Programming FPGA in Haskell'''(Talk) || ''Alexander Thiemann'', '''Spock - Powerful Elegant Web Applications''' (Talk) ||<br />
|- <br />
| 7pm || '''BBQ''' at lake (depends on weather, otherwise Sa) || '''BBQ''' at lake (depends on weather, otherwise Fr) ||<br />
|}<br />
<br />
== Talks ==<br />
<br />
The links to the slides of the talks can be found on the [https://github.com/meiersi/HaskellerZ#2016-meetups '''HaskellerZ github page'''].<br />
<br />
=== Alexander Thiemann ===<br />
<br />
Title: Spock - Powerful Elegant Web Applications<br />
<br />
Abstract: This talk will give an insight on web development using the Haskell web framework «Spock» ( https://www.spock.li ). It will give a short overview of the possibilities like type-safe routing, hyperlinks, sessions, database queries, templates and JSON parsing/generation and then we will also look at a technique to build a full stack application with Spock and GHCJS as used in production in TramCloud.<br />
<br />
=== Andres Löh ===<br />
<br />
Title: Generic (and type-level) programming with generics-sop<br />
<br />
Abstract: Many Haskell functions can be defined for a large class of datatypes in<br />
a systematic way. Examples include structural equality and comparisons,<br />
all kind of (de)serialization functions (plain text, JSON, binary,<br />
etc.), traversal and access functions such as lenses and their various<br />
variants, functions for querying and updating databases and many more.<br />
For some built-in type classes, Haskell offers the "deriving" construct,<br />
but GHC also has a "generics" feature that allows users to define their<br />
own derivable classes, without having to resort to Template Haskell.<br />
<br />
In this talk, I'm going to introduce "generics-sop", a library that<br />
supports such datatype-generic programming by means of a number of<br />
powerful, higher-order combinators that can be reused and composed in<br />
several ways. This approach makes use of several type-system extensions<br />
available for GHC, such as GADTs (in particular, for heterogeneous<br />
lists), data kinds, kind polymorphism, constraint kinds, and rank-n<br />
polymorphism. We will start from examples and see how we can define<br />
a few generic functions using generics-sop and then, as far as time<br />
permits, look more closely at how it all works.<br />
<br />
=== Bas van Dijk ===<br />
<br />
Title: Functional Programming at LumiGuide<br />
<br />
Abstract: tba<br />
<br />
=== David Luposchainsky ===<br />
<br />
Title: Low-level Haskell: an interactive tour through the STG<br />
<br />
Abstract: Executing lazy functional programs is often met with suspicion, as something processors somehow don't do “naturally”. In this talk, I will show just how wrong such statements are.<br />
<br />
We will take a tour through a few well-known Haskell programs in their STG representation. STG is a tiny functional language used in GHC's compiler backend. What makes STG particularly interesting is it is feels close to both Haskell and the metal, and allows us to watch how Haskell programs are executed step-by-step.<br />
<br />
Why does `foldl (+) 0` overflow, and is it the stack or the heap? How are things pushed onto the stack or allocated on the heap anyway, and what cleans them up again? How does laziness work? Why are unboxed values often faster, how does a list look like in memory, how efficient is the popular Quicksort-inspired example algorithm really?<br />
<br />
The goal of the talk is to introduce the audience into the operational semantics of Haskell, by executing a couple of small programs everyone should be familiar with, and discussing what happens during execution. The topic is suitable for audiences of all skill levels.<br />
<br />
=== Edward Kmett ===<br />
<br />
Title: tba<br />
<br />
Abstract: tba<br />
<br />
=== Francesco Mazzoli ===<br />
<br />
Title: Parallelizing and distributing scientific software in Haskell<br />
<br />
Abstract: Suppose you have a pure, CPU intensive function that you need to run on 10000 different inputs. This seems the dream task to parallelize in Haskell, but as it turns out it is not as easy as one might think. In this talk I'll describe what we needed to do to parallelize such an application on machines with as many as 18 cores, and further what we needed to do to distributed the same application across separate machines, on hundreds of cores.<br />
<br />
=== Luka Rahne ===<br />
<br />
Title: Clash - Programming FPGA in Haskell<br />
<br />
Abstract: Clash is compiler and programming language, that transform Haskell like<br />
language into description of digital circuit aiming for both FPGA and<br />
ASIC. Tools from Haskell ecosystem including libraries and GHC features<br />
can be used to make reliable and efficient hardware. Basics of FPGA and<br />
digital circuits design will be presented and also how Clash can be used<br />
to describe, simulate and synthesize such circuits.<br />
<br />
== Projects ==<br />
See the [[/Projects|projects page]].<br />
<br />
== Communication ==<br />
If you have any questions before the event, please reach out to Alexander Bernauer <acopton@gmail.com>.<br />
<br />
Before the event, and in particular during the event, you can find us and other participants on irc://freenode.net/#zurihac. In addition, feel free to post on Twitter and Google+ using the hashtag '''#ZuriHac2016'''.<br />
<br />
== Organizers ==<br />
The events are organized in name of the [http://www.meetup.com/HaskellerZ/ Zurich Haskell meetup group] by the following people:<br />
<br />
* Gleb Peregud (Google)<br />
* Alexander Bernauer (Digital Asset)<br />
* Simon Meier (Digital Asset)<br />
* Ivan Kristo (Google)<br />
* Jasper Van der Jeugt (Luminal)<br />
* Arvin Moezzi (Google)<br />
<br />
[[Category:Community]]<br />
[[Category:Events]]<br />
[[Category:Hackathon]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Language/HaskellPrime&diff=60748Language/HaskellPrime2016-05-06T14:03:14Z<p>Blackout: /* Members */ Add affiliation for quchen/David Luposchainsky</p>
<hr />
<div>= Haskell Prime Committee =<br />
<br />
This page lists information about the goings-on of the Haskell Prime Committee, which is attempting to make a new Haskell 2020 standard.<br />
<br />
=== Members ===<br />
<br />
The members of the committee are as follows:<br />
<br />
* Andres Löh<br />
* Antonio Nikishaev<br />
* Austin Seipp<br />
* Carlos Camarao de Figueiredo<br />
* Carter Schonwald<br />
* David Luposchainsky (TNG Technology Consulting, dluposchainsky(λ)googlesemail, a.k.a. quchen)<br />
* Henk-Jan van Tuyl<br />
* Henrik Nilsson<br />
* Herbert Valerio Riedel<br />
* Iavor Diatchki<br />
* John Wiegley<br />
* José Manuel Calderón Trilla (Galois, Inc.; jmct@jmct.cc; a.k.a. jmct)<br />
* Jurriaan Hage<br />
* Lennart Augustsson (Standard Chartered Bank, lennart@augustsson.net, a.k.a. augustss)<br />
* M Farkas-Dyck<br />
* Mario Blažević (Stilo International plc; mblazevic@stilo.com)<br />
* Nicolas Wu<br />
* Richard Eisenberg (University of Pennsylvania; eir@cis.upenn.edu; a.k.a. goldfire)<br />
* Vitaly Bragilevsky (Southern Federal University, Rostov-on-Don, Russia; bravit@sfedu.ru; a.k.a. bravit)<br />
* Wren Romano</div>Blackouthttps://wiki.haskell.org/index.php?title=Language/HaskellPrime&diff=60719Language/HaskellPrime2016-04-29T18:45:26Z<p>Blackout: David's details added</p>
<hr />
<div>= Haskell Prime Committee =<br />
<br />
This page lists information about the goings-on of the Haskell Prime Committee, which is attempting to make a new Haskell 2020 standard.<br />
<br />
=== Members ===<br />
<br />
The members of the committee are as follows:<br />
<br />
* Andres Löh<br />
* Antonio Nikishaev<br />
* Austin Seipp<br />
* Carlos Camarao de Figueiredo<br />
* Carter Schonwald<br />
* David Luposchainsky (dluposchainsky(λ)googlesemail, aka quchen)<br />
* Henk-Jan van Tuyl<br />
* Henrik Nilsson<br />
* Herbert Valerio Riedel<br />
* Iavor Diatchki<br />
* John Wiegley<br />
* José Manuel Calderón Trilla (Galois, Inc.; jmct@jmct.cc; a.k.a. jmct)<br />
* Jurriaan Hage<br />
* Lennart Augustsson<br />
* M Farkas-Dyck<br />
* Mario Blažević<br />
* Nicolas Wu<br />
* Richard Eisenberg (University of Pennsylvania; eir@cis.upenn.edu; a.k.a. goldfire)<br />
* Vitaly Bragilevsky (Southern Federal University, Rostov-on-Don, Russia; bravit@sfedu.ru; a.k.a. bravit)<br />
* Wren Romano</div>Blackouthttps://wiki.haskell.org/index.php?title=MonadFail_Proposal&diff=60384MonadFail Proposal2015-11-25T18:00:10Z<p>Blackout: Add note that article was moved to Haskell' Wiki</p>
<hr />
<div>'''Note:''' this proposal page has been moved to [https://prime.haskell.org/wiki/Libraries/Proposals/MonadFail the Haskell Prime Wiki]; the article below is unmaintained.<br />
<br />
<br />
== <code>MonadFail</code> proposal (MFP) ==<br />
<br />
A couple of years ago, we proposed to make <code>Applicative</code> a superclass of <code>Monad</code> which successfully killed the single most ugly thing in Haskell as of GHC 7.10.<br />
<br />
Now, it's time to tackle the other major issue with <code>Monad</code> <code>fail</code> being a part of it.<br />
<br />
You can contact me as usual via IRC/Freenode as ''quchen'', or by email to ''dluposchainsky at the email service of Google''. This file was posted on the ghc-devs@ and [http://permalink.gmane.org/gmane.comp.lang.haskell.libraries/24910 libraries@ mailing lists], as well as on [http://www.reddit.com/r/haskell/comments/397k1a/monadfail_proposal_mfp_moving_fail_out_of_monad/ Reddit].<br />
<br />
This proposal was first posted on [https://github.com/quchen/articles/blob/master/monad_fail.md quchen's articles Github repo].<br />
<br />
=== Overview ===<br />
<br />
* '''The problem''' - reason for the proposal<br />
* '''MonadFail class''' - the solution<br />
* '''Discussion''' - explaining our design choices<br />
* '''Adapting old code''' - how to prepare current code to transition smoothly<br />
* '''Estimating the breakage''' - how much stuff we will break<br />
* '''Transitional strategy''' - how to break as little as possible while transitioning<br />
* '''Current status'''<br />
<br />
=== The problem ===<br />
<br />
Currently, the <code><-</code> symbol is unconditionally desugared as follows:<br />
<br />
<syntaxhighlight lang="haskell"><br />
do pat <- computation >>> let f pat = more<br />
more >>> f _ = fail "..."<br />
>>> in computation >>= f<br />
</syntaxhighlight><br />
<br />
The problem with this is that <code>fail</code> cannot (!) be sensibly implemented for many monads, for example <code>Either</code>, <code>State</code>, <code>IO</code>, and <code>Reader</code>. In those cases it defaults to <code>error</code> As a consequence, in current Haskell, you can not use <code>Monad</code> polymorphic code safely, because although it claims to work for all <code>Monad</code> , it might just crash on you. This kind of implicit non-totality baked into the class is ''terrible''.<br />
<br />
The goal of this proposal is adding the <code>fail</code> only when necessary and reflecting that in the type signature of the <code>do</code> block, so that it can be used safely, and more importantly, is guaranteed not to be used if the type signature does not say so.<br />
<br />
=== <code>MonadFail</code> class ===<br />
<br />
To fix this, introduce a new typeclass:<br />
<br />
<syntaxhighlight lang="haskell"><br />
class Monad m => MonadFail m where<br />
fail :: String -> m a<br />
</syntaxhighlight><br />
<br />
Desugaring can now be changed to produce this constraint when necessary. For this, we have to decide when a pattern match can not fail; if this is the case, we can omit inserting the <code>fail</code> call.<br />
<br />
The most trivial examples of unfailable patterns are of course those that match anywhere unconditionally,<br />
<br />
<syntaxhighlight lang="haskell"><br />
do x <- action >>> let f x = more<br />
more >>> in action >>= f<br />
</syntaxhighlight><br />
<br />
In particular, the programmer can assert any pattern be unfailable by making it irrefutable using a prefix tilde:<br />
<br />
<syntaxhighlight lang="haskell"><br />
do ~pat <- action >>> let f ~pat = more<br />
more >>> in action >>= f<br />
</syntaxhighlight><br />
<br />
A class of patterns that are conditionally failable are <code>newtype</code> , and single constructor <code>data</code> types, which are unfailable by themselves, but may fail if matching on their fields is done with failable patterns.<br />
<br />
<syntaxhighlight lang="haskell"><br />
data Newtype a = Newtype a<br />
<br />
-- "x" cannot fail<br />
do Newtype x <- action >>> let f (Newtype x) = more<br />
more >>> in action >>= f<br />
<br />
-- "Just x" can fail<br />
do Newtype (Just x) <- action >>> let f (Newtype (Just x)) = more<br />
more >>> f _ = fail "..."<br />
>>> in action >>= f<br />
</syntaxhighlight><br />
<br />
<code>ViewPatterns</code> are as failable as the pattern the view is matched against. Patterns like <code>(Just -> Just x)</code> should generate a <code>MonadFail</code> constraint even when it's "obvious" from the view's implementation that the pattern will always match. From an implementor's perspective, this means that only types (and their constructors) have to be looked at, not arbitrary values (like functions), which is impossible to do statically in general.<br />
<br />
<syntaxhighlight lang="haskell"><br />
do (view -> pat) <- action >>> let f (view -> pat) = more<br />
more >>> f _ = fail "..."<br />
>>> in action >>= f<br />
<br />
do (view -> ~pat) <- action >>> let f (view -> ~pat) = more<br />
more >>> in action >>= f<br />
</syntaxhighlight><br />
<br />
[Edward Kmett: `(view -> pat)` should be unfailing iff pat is unfailing.]<br />
<br />
A similar issue arises for <code>PatternSynonyms</code> which we cannot inspect during compilation sufficiently. A pattern synonym will therefore always be considered failable.<br />
<br />
<syntaxhighlight lang="haskell"><br />
do PatternSynonym x <- action >>> let f PatternSynonym x = more<br />
more >>> f _ = fail "..."<br />
>>> in action >>= f<br />
</syntaxhighlight><br />
<br />
[Edward Kmett: We have the contents of the pattern synonym available to us at the definition site. With some work we should be able to expose it enough that the compiler can see through it:<br />
<br />
<syntaxhighlight lang="haskell"><br />
pattern Foo a b = Bar a 0 b<br />
pattern Baz a b c <- Quux a b c<br />
</syntaxhighlight><br />
<br />
Both of those tell us the "real" desugaring as just another pattern we could recurse into.]<br />
<br />
=== Discussion ===<br />
<br />
* What laws should <code>fail</code> follow?<br />
** '''Left zero''': <code>∀ s f. fail s >>= f ≡ fail s</code>.<br />
** '''Right zero''': <code>∀ v s. v >> fail s ≡ fail s</code>.<br />
* What is the relationship to <code>MonadPlus</code>?<br />
** As the laws above indicate, <code>fail</code> is a close relative of <code>mzero</code>. We could suggest a default definition of <code>fail _ = mzero</code>, which shows the intended usage and effect of the <code>MonadFail</code> class.<br />
** However, we should not remove <code>fail</code> and use only <code>mzero</code> instead. <br />
*** Not all types with <code>Monad</code> instances have <code>MonadPlus</code> instances.<br />
*** Some types do use the <code>String</code> argument to <code>fail</code>. For example, a parser might fail with a message involving positional information. <code>Binary</code> uses <code>fail</code> as their only interface to fail a decoding step.<br />
*** Some types have different definitions for <code>mzero</code> and <code>fail</code>. Although <code>STM</code> is <code>MonadPlus</code> it uses the default <code>fail = error</code>. It should therefore not get a <code>MonadFail</code> instance.<br />
* Rename <code>fail</code>?<br />
** '''No.''' Old code might use <code>fail</code> explicitly and we should avoid breaking it. The Report talks about <code>fail</code> and we have a solid migration strategy that does not require a renaming.<br />
* Remove the <code>String</code> argument?<br />
** '''No.''' The <code>String</code> might help error reporting and debugging. <code>String</code> may be ugly, but it's the de facto standard for simple text in GHC. No high performance string operations are to be expected with <code>fail</code> so this breaking change would in no way be justified. Also note that explicit <code>fail</code> calls would break if we removed the argument.<br />
* How sensitive would existing code be to subtle changes in the strictness behaviour of <code>do</code> notation pattern matching?<br />
** '''It doesn't.''' The implementation does not affect strictness at all, only the desugaring step. Care must be taken when fixing warnings by making patterns irrefutable using <code>~</code> as that ''does'' affect strictness. (Cf. difference between lazy/strict State)<br />
* Do we need a class constraint (e.g. <code>Monad</code>) on <code>MonadFail</code>?<br />
** '''Yes.''' The intended use of <code>fail</code> is for desugaring <code>do</code>-notation, not generally for any <code>String -> m a</code> function. Given that goal, we would rather keep the constraints simple as <code>MonadFail m =></code> rather than the somewhat redundant <code>(Monad m, MonadFail m) =></code>.<br />
* Can we relax the class constraint from <code>Monad</code> to <code>Applicative</code>?<br />
** We don't necessarily have to choose now. Since <code>Applicative</code> is a superclass of <code>Monad</code>, it is possible to change the superclass for <code>MonadFail</code> to <code>Applicative</code> later. This will naturally require a migration period, and the name will, of course, become misleading.<br />
** For the sake of discussion, let's use the following definition:<syntaxhighlight lang="haskell">class Applicative f => ApplicativeFail f where fail :: String -> f a</syntaxhighlight><br />
** '''Pros'''<br />
*** <code>ApplicativeDo</code> is coming, and <code>fail</code> may be useful to combine pattern matching and <code>Applicative</code> code.<br />
*** If the <code>Monad</code> constraint is kept, that would force <code>Applicative</code> code with pattern matching to be <code>Monad</code> code.<br />
** '''Cons'''<br />
*** The constraints for <code>Monad</code> code using <code>fail</code> become <code>(Monad m, ApplicativeFail m) =></code> instead of the simpler <code>MonadFail m =></code>. If we expect the common use of <code>fail</code> to be in <code>Monad</code> — not <code>Applicative</code> — <code>do</code>-notation, this leaves us with more verbose constraints.<br />
** Here are alternative definitions (with names open to debate) that would allow us to keep the constraints simple:<br />
*** <syntaxhighlight lang="haskell">class Applicative f => ApplicativeFail f where failA :: String -> f a</syntaxhighlight><br />
*** <syntaxhighlight lang="haskell">class ApplicativeFail m => MonadFail m where fail :: String -> m a; fail = failA</syntaxhighlight><br />
*** Since we do not have much experience using <code>ApplicativeDo</code>, it is not yet clear that this large of a change is useful.<br />
* Which types with <code>Monad</code> instances will not have <code>MonadFail</code> instances?<br />
** <code>base</code>: <code>Either</code><br />
** <code>transformers</code>:<br />
** <code>stm</code>: <code>STM</code><br />
* What <code>MonadFail</code> instances will be created?<br />
** <code>base</code>: <code>IO</code><br />
** <code>transformers</code>:<br />
*** Proposal for an <code>Either</code> instance using <code>Monad</code> instance in <code>Control.Monad.Trans.Error</code>:<syntaxhighlight lang="haskell">instance MonadFail (Either String) where fail = Left </syntaxhighlight><br />
<br />
=== Adapting old code ===<br />
<br />
<ul><br />
<li>Help! My code is broken because of a missing <code>MonadFail</code> instance! ''Here are your options:''<br />
<ol><li>Write a <code>MonadFail</code> instance (and bring it into scope)<br />
<syntaxhighlight lang="haskell"><br />
#if !MIN_VERSION_base(4,11,0)<br />
-- Control.Monad.Fail import will become redundant in GHC 7.16+<br />
import qualified Control.Monad.Fail as Fail<br />
#endif<br />
import Control.Monad<br />
<br />
instance Monad Foo where<br />
(>>=) = <...bind impl...><br />
-- NB: <code>return</code> defaults to <code>pure</code><br />
#if !MIN_VERSION_base(4,11,0)<br />
-- Monad(fail) will be removed in GHC 7.16+<br />
fail = Fail.fail<br />
#endif<br />
<br />
instance MonadFail Foo where<br />
fail = <...fail implementation...><br />
</syntaxhighlight></li><br />
<li>Change your pattern to be irrefutable</li><br />
<li>Emulate the old behaviour by desugaring the pattern match by hand:<br />
<syntaxhighlight lang="haskell"><br />
do Left e <- foobar<br />
stuff<br />
</syntaxhighlight> becomes <syntaxhighlight lang="haskell"><br />
do x <- foobar<br />
e <- case x of<br />
Left e' -> e'<br />
Right r -> error "Pattern match failed" -- Boooo<br />
stuff<br />
</syntaxhighlight><br />
The point is you'll have to do your dirty laundry yourself now if you have a value that ''you'' know will always match, and if you don't handle the other patterns you'll get incompleteness warnings, and the compiler won't silently eat those for you.</li><br />
</ol></li><br />
<li>Help! My code is broken because you removed <code>fail</code> from <code>Monad</code> but my class defines it! ''Delete that part of the instance definition.''</li></ul><br />
<br />
=== Esimating the breakage ===<br />
<br />
Using our initial implementation, I compiled stackage-nightly, and grepped the logs for the warnings. Assuming my implementation is correct, the number of "missing <code>MonadFail</code> warnings generated is 487. Note that I filtered out <code>[]</code> <code>Maybe</code> and <code>ReadPrec</code> since those can be given a <code>MonadFail</code> instance from within GHC, and no breakage is expected from them.<br />
The build logs can be found [https://www.dropbox.com/s/knz0i979skam4zs/stackage-build.tar.xz?dl=0 here]. Search for "failable pattern" to find your way to the still pretty raw warnings.<br />
<br />
Here are some commands you might find interesting for exploring the logs:<br />
<br />
<syntaxhighlight lang="bash"><br />
# List all packages generating warnings (57 of them)<br />
grep "is used in the context" '' | \<br />
grep -v '(‘\[|Maybe|ReadPrec)' | \<br />
perl -pe 's#^(.'')\.log.''$#\1#' | \<br />
uniq -u<br />
<br />
# Histogram of the breaking contexts (mostly IO and parsers)<br />
grep "is used in the context" '' | \<br />
grep -v '(‘\[|Maybe|ReadPrec)' | \<br />
perl -pe 's#^.''in the context ‘([^ ]+).''$#\1#' | \<br />
sort | \<br />
uniq -c | \<br />
sort -rg<br />
</syntaxhighlight><br />
<br />
=== Transitional strategy ===<br />
<br />
The roadmap is similar to the [https://github.com/quchen/articles/blob/master/applicative_monad.md AMP], the main difference being that since <code>MonadFail</code> does not exist yet, we have to introduce new functionality and then switch to it.<br />
<br />
<ol><br />
<li>GHC 8.0 / base-4.9<br />
<ul><br />
<li>Add module <code>Control.Monad.Fail</code> with new class <code>MonadFail(fail)</code> so people can start writing instances for it. <code>Control.Monad</code> only re-exports the class <code>MonadFail</code> but not its <code>fail</code> method. NB: At this point, <code>Control.Monad.Fail.fail</code> clashes with <code>Prelude.fail</code> and <code>Control.Monad.fail</code>.</li><br />
<li>Add a language extension <code>-XMonadFailDesugaring</code> that changes desugaring to use <code>MonadFail(fail)</code> instead of <code>Monad(fail)</code> This has the effect that typechecking will infer a <code>MonadFail</code> constraint for <code>do</code> blocks with failable patterns, just as it is planned to do when the entire thing is done.</li><br />
<li>Add a warning when a <code>do</code> block that contains a failable pattern is desugared, but there is no <code>MonadFail</code> instance in scope: "Please add the instance or change your pattern matching." Add a flag to control whether this warning appears, but leave it off by default.</li><br />
<li>Add a warning when an instance implements the <code>fail</code> function (or when <code>fail</code> is imported as a method of <code>Monad</code> , as it will be removed from the <code>Monad</code> class in the future. (See also [https://ghc.haskell.org/trac/ghc/ticket/10071 GHC #10071]). Leave it off by default.</li><br />
</ul><br />
</li><br />
<li>GHC 8.4<br />
<ul><br />
<li>Turn on the warning about missing <code>MonadFail</code> instances that we added in 8.0 by default.</li><br />
</ul><br />
</li><br />
<li>GHC 8.6<br />
<ul><br />
<li>Switch <code>-XMonadFailDesugaring</code> on by default.</li><br />
<li>Warnings are still issued if the desugaring extension has been explicitly disabled.</li><br />
<li>Turn on the warning about explicit definition of `fail` in Monad that we added in 8.0 by default.</li><br />
</ul><br />
</li><br />
<li>GHC 8.8<br />
<ul><br />
<li>Remove <code>-XMonadFail</code> leaving its effects on at all times.</li><br />
<li>Remove <code>fail</code> from <code>Monad</code></li><br />
<li>Instead, re-export <code>Control.Monad.Fail.fail</code> as <code>Prelude.fail</code> and <code>Control.Monad.fail</code></li><br />
<li><code>Control.Monad.Fail</code> is now a redundant module that can be considered deprecated.</li><br />
</ul><br />
</li><br />
</ol><br />
<br />
=== Current status ===<br />
<br />
* [https://wiki.haskell.org/ZuriHac2015 ZuriHac 2015 (29.5. - 31.5.)]: Franz Thoma (@fmthoma) and me (David Luposchainsky aka @quchen) started implementing the MFP in GHC.<br />
** Desugaring to the new <code>fail</code> can be controlled via a new language extension, <code>MonadFailDesugaring</code><br />
** If the language extension is turned off, a warning will be emitted for code that would break if it was enabled.<br />
** Warnings are emitted for types that ''have'' a ''MonadFail'' instance. This still needs to be fixed.<br />
** The error messages are readable, but should be more so. We're still on this.<br />
* 2015-06-09: Estimated breakage by compiling Stackage. Smaller than expected.<br />
* 2015-06-09 (late): Published. People seem to like the idea, with a couple of pain points remaining.<br />
* 2015-06-16: [https://github.com/quchen/articles/blob/master/monad_fail_update1.md Update 1 posted.]<br />
* 2015-09-18: [https://phabricator.haskell.org/D1248 Patch nearly finished. Some nontrivial tests still fail.]</div>Blackouthttps://wiki.haskell.org/index.php?title=MonadFail_Proposal&diff=60136MonadFail Proposal2015-09-18T20:06:22Z<p>Blackout: Initial translation from my Github-based Markdown article</p>
<hr />
<div>== <code>MonadFail</code> proposal (MFP) ==<br />
<br />
A couple of years ago, we proposed to make <code>Applicative</code> a superclass of <code>Monad</code> which successfully killed the single most ugly thing in Haskell as of GHC 7.10.<br />
<br />
Now, it's time to tackle the other major issue with <code>Monad</code> <code>fail</code> being a part of it.<br />
<br />
You can contact me as usual via IRC/Freenode as ''quchen'', or by email to ''dluposchainsky at the email service of Google''. This file was posted on the ghc-devs@ and [http://permalink.gmane.org/gmane.comp.lang.haskell.libraries/24910 libraries@ mailing lists], as well as on [http://www.reddit.com/r/haskell/comments/397k1a/monadfail_proposal_mfp_moving_fail_out_of_monad/ Reddit].<br />
<br />
This proposal was first posted on [https://github.com/quchen/articles/blob/master/monad_fail.md quchen's articles Github repo].<br />
<br />
=== Overview ===<br />
<br />
* '''The problem''' - reason for the proposal<br />
* '''MonadFail class''' - the solution<br />
* '''Discussion''' - explaining our design choices<br />
* '''Adapting old code''' - how to prepare current code to transition smoothly<br />
* '''Estimating the breakage''' - how much stuff we will break<br />
* '''Transitional strategy''' - how to break as little as possible while transitioning<br />
* '''Current status'''<br />
<br />
=== The problem ===<br />
<br />
Currently, the <code><-</code> symbol is unconditionally desugared as follows:<br />
<br />
<syntaxhighlight lang="haskell"><br />
do pat <- computation >>> let f pat = more<br />
more >>> f _ = fail "..."<br />
>>> in computation >>= f<br />
</syntaxhighlight><br />
<br />
The problem with this is that <code>fail</code> cannot (!) be sensibly implemented for many monads, for example <code>State</code> <code>IO</code> <code>Reader</code> In those cases it defaults to <code>error</code> As a consequence, in current Haskell, you can not use <code>Monad</code> polymorphic code safely, because although it claims to work for all <code>Monad</code> , it might just crash on you. This kind of implicit non-totality baked into the class is ''terrible''.<br />
<br />
The goal of this proposal is adding the <code>fail</code> only when necessary and reflecting that in the type signature of the <code>do</code> block, so that it can be used safely, and more importantly, is guaranteed not to be used if the type signature does not say so.<br />
<br />
=== <code>MonadFail</code> class ===<br />
<br />
To fix this, introduce a new typeclass:<br />
<br />
<syntaxhighlight lang="haskell"><br />
class Monad m => MonadFail m where<br />
fail :: String -> m a<br />
</syntaxhighlight><br />
<br />
Desugaring can now be changed to produce this constraint when necessary. For this, we have to decide when a pattern match can not fail; if this is the case, we can omit inserting the <code>fail</code> call.<br />
<br />
The most trivial examples of unfailable patterns are of course those that match anywhere unconditionally,<br />
<br />
<syntaxhighlight lang="haskell"><br />
do x <- action >>> let f x = more<br />
more >>> in action >>= f<br />
</syntaxhighlight><br />
<br />
In particular, the programmer can assert any pattern be unfailable by making it irrefutable using a prefix tilde:<br />
<br />
<syntaxhighlight lang="haskell"><br />
do ~pat <- action >>> let f ~pat = more<br />
more >>> in action >>= f<br />
</syntaxhighlight><br />
<br />
A class of patterns that are conditionally failable are <code>newtype</code> , and single constructor <code>data</code> types, which are unfailable by themselves, but may fail if matching on their fields is done with failable patterns.<br />
<br />
<syntaxhighlight lang="haskell"><br />
data Newtype a = Newtype a<br />
<br />
-- "x" cannot fail<br />
do Newtype x <- action >>> let f (Newtype x) = more<br />
more >>> in action >>= f<br />
<br />
-- "Just x" can fail<br />
do Newtype (Just x) <- action >>> let f (Newtype (Just x)) = more<br />
more >>> f _ = fail "..."<br />
>>> in action >>= f<br />
</syntaxhighlight><br />
<br />
<code>ViewPatterns</code> are as failable as the pattern the view is matched against. Patterns like <code>(Just -> Just x)</code> should generate a <code>MonadFail</code> constraint even when it's "obvious" from the view's implementation that the pattern will always match. From an implementor's perspective, this means that only types (and their constructors) have to be looked at, not arbitrary values (like functions), which is impossible to do statically in general.<br />
<br />
<syntaxhighlight lang="haskell"><br />
do (view -> pat) <- action >>> let f (view -> pat) = more<br />
more >>> f _ = fail "..."<br />
>>> in action >>= f<br />
<br />
do (view -> ~pat) <- action >>> let f (view -> ~pat) = more<br />
more >>> in action >>= f<br />
</syntaxhighlight><br />
<br />
A similar issue arises for <code>PatternSynonyms</code> which we cannot inspect during compilation sufficiently. A pattern synonym will therefore always be considered failable.<br />
<br />
<syntaxhighlight lang="haskell"><br />
do PatternSynonym x <- action >>> let f PatternSynonym x = more<br />
more >>> f _ = fail "..."<br />
>>> in action >>= f<br />
</syntaxhighlight><br />
<br />
=== Discussion ===<br />
<br />
* Although for many <code>MonadPlus</code> <code>fail _ = mzero</code> a separate <code>MonadFail</code> class should be created instead of just using that.<br />
** A parser might fail with an error message involving positional information. Some libraries, like <code>Binary</code> provide <code>fail</code> as their only interface to fail a decoding step.<br />
** Although <code>STM</code> is <code>MonadPlus</code> it uses the default <code>fail = error</code> It will therefore not get a <code>MonadFail</code> instance.<br />
* What laws should <code>fail</code> follow? '''Left zero''', <syntaxhighlight lang="haskell"> ∀ s f. fail s >>= f ≡ fail s</syntaxhighlight> A call to <code>fail</code> should abort the computation. In this sense, <code>fail</code> would become a close relative of <code>mzero</code> It would work well with the common definition <code>fail _ = mzero</code> and give a simple guideline to the intended usage and effect of the <code>MonadFail</code> class.<br />
* Rename <code>fail</code> '''No.''' Old code might use <code>fail</code> explicitly and we might avoid breaking it, the Report talks about <code>fail</code> and we have a solid migration strategy that does not require a renaming.<br />
* Remove the <code>String</code> argument? '''No.''' The <code>String</code> might help error reporting and debugging. <code>String</code> may be ugly, but it's the de facto standard for simple text in GHC. No high performance string operations are to be expected with <code>fail</code> so this breaking change would in no way be justified. Also note that explicit <code>fail</code> calls would break if we removed the argument.<br />
* How sensitive would existing code be to subtle changes in the strictness behaviour of <code>do</code> notation pattern matching? '''It doesn't.''' The implementation does not affect strictness at all, only the desugaring step. Care must be taken when fixing warnings by making patterns irrefutable using <code>~</code> as that ''does'' affect strictness. (Cf. difference between lazy/strict State)<br />
* The <code>Monad</code> constraint for <code>MonadFail</code> seems unnecessary. Should we drop or relax it? What other things should be considered?<br />
** Applicative <code>do</code> notation is coming sooner or later, <code>fail</code> might be useful in this more general scenario. Due to the AMP, it is trivial to change the <code>MonadFail</code> superclass to <code>Applicative</code> later. (The name will be a bit misleading, but it's a very small price to pay.)<br />
** The class might be misused for a strange pointed type if left without any constraint. This is not the intended use at all. I think we should keep the <code>Monad</code> superclass for three main reasons:<br />
** We don't want to see <code>(Monad m, MonadFail m) =></code> all over the place.<br />
** The primary intended use of <code>fail</code> is for desugaring do-notation anyway.<br />
** Retroactively removing superclasses is easy, but adding them is hard (see AMP).<br />
<br />
=== Adapting old code ===<br />
<br />
<ul><br />
<li>Help! My code is broken because of a missing <code>MonadFail</code> instance! ''Here are your options:''<br />
<ol><li>Write a <code>MonadFail</code> instance (and bring it into scope)<br />
<syntaxhighlight lang="haskell"><br />
#if !MIN_VERSION_base(4,11,0)<br />
-- Control.Monad.Fail import will become redundant in GHC 7.16+<br />
import qualified Control.Monad.Fail as Fail<br />
#endif<br />
import Control.Monad<br />
<br />
instance Monad Foo where<br />
(>>=) = <...bind impl...><br />
-- NB: <code>return</code> defaults to <code>pure</code><br />
#if !MIN_VERSION_base(4,11,0)<br />
-- Monad(fail) will be removed in GHC 7.16+<br />
fail = Fail.fail<br />
#endif<br />
<br />
instance MonadFail Foo where<br />
fail = <...fail implementation...><br />
</syntaxhighlight></li><br />
<li>Change your pattern to be irrefutable</li><br />
<li>Emulate the old behaviour by desugaring the pattern match by hand:<br />
<syntaxhighlight lang="haskell"><br />
do Left e <- foobar<br />
stuff<br />
</syntaxhighlight> becomes <syntaxhighlight lang="haskell"><br />
do x <- foobar<br />
e <- case x of<br />
Left e' -> e'<br />
Right r -> error "Pattern match failed" -- Boooo<br />
stuff<br />
</syntaxhighlight><br />
The point is you'll have to do your dirty laundry yourself now if you have a value that ''you'' know will always match, and if you don't handle the other patterns you'll get incompleteness warnings, and the compiler won't silently eat those for you.</li><br />
</ol></li><br />
<li>Help! My code is broken because you removed <code>fail</code> from <code>Monad</code> but my class defines it! ''Delete that part of the instance definition.''</li></ul><br />
<br />
=== Esimating the breakage ===<br />
<br />
Using our initial implementation, I compiled stackage-nightly, and grepped the logs for the warnings. Assuming my implementation is correct, the number of "missing <code>MonadFail</code> warnings generated is 487. Note that I filtered out <code>[]</code> <code>Maybe</code> and <code>ReadPrec</code> since those can be given a <code>MonadFail</code> instance from within GHC, and no breakage is expected from them.<br />
The build logs can be found [https://www.dropbox.com/s/knz0i979skam4zs/stackage-build.tar.xz?dl=0 here]. Search for "failable pattern" to find your way to the still pretty raw warnings.<br />
<br />
Here are some commands you might find interesting for exploring the logs:<br />
<br />
<syntaxhighlight lang="bash"><br />
# List all packages generating warnings (57 of them)<br />
grep "is used in the context" '' | \<br />
grep -v '(‘\[|Maybe|ReadPrec)' | \<br />
perl -pe 's#^(.'')\.log.''$#\1#' | \<br />
uniq -u<br />
<br />
# Histogram of the breaking contexts (mostly IO and parsers)<br />
grep "is used in the context" '' | \<br />
grep -v '(‘\[|Maybe|ReadPrec)' | \<br />
perl -pe 's#^.''in the context ‘([^ ]+).''$#\1#' | \<br />
sort | \<br />
uniq -c | \<br />
sort -rg<br />
</syntaxhighlight><br />
<br />
=== Transitional strategy ===<br />
<br />
The roadmap is similar to the [https://github.com/quchen/articles/blob/master/applicative_monad.md AMP], the main difference being that since <code>MonadFail</code> does not exist yet, we have to introduce new functionality and then switch to it.<br />
<br />
<ol><br />
<li>GHC 8.0 / base-4.9<br />
<ul><br />
<li>Add module <code>Control.Monad.Fail</code> with new class <code>MonadFail(fail)</code> so people can start writing instances for it. <code>Control.Monad</code> only re-exports the class <code>MonadFail</code> but not its <code>fail</code> method. NB: At this point, <code>Control.Monad.Fail.fail</code> clashes with <code>Prelude.fail</code> and <code>Control.Monad.fail</code>.</li><br />
<li>Add a language extension <code>-XMonadFailDesugaring</code> that changes desugaring to use <code>MonadFail(fail)</code> instead of <code>Monad(fail)</code> This has the effect that typechecking will infer a <code>MonadFail</code> constraint for <code>do</code> blocks with failable patterns, just as it is planned to do when the entire thing is done.</li><br />
<li>Warn when a <code>do</code> block that contains a failable pattern is desugared, but there is no <code>MonadFail</code> instance in scope: "Please add the instance or change your pattern matching." Add a flag to control whether this warning appears.</li><br />
<li>Warn when an instance implements the <code>fail</code> function (or when <code>fail</code> is imported as a method of <code>Monad</code> , as it will be removed from the <code>Monad</code> class in the future. (See also [https://ghc.haskell.org/trac/ghc/ticket/10071 GHC #10071])</li><br />
</ul><br />
</li><br />
<li>GHC 8.2<br />
<ul><br />
<li>Switch <code>-XMonadFailDesugaring</code> on by default.</li><br />
<li>Warnings are still issued if the desugaring extension has been explicitly disabled.</li><br />
</ul><br />
</li><br />
<li>GHC 8.4<br />
<ul><br />
<li>Remove <code>-XMonadFail</code> leaving its effects on at all times.</li><br />
<li>Remove <code>fail</code> from <code>Monad</code></li><br />
<li>Instead, re-export <code>Control.Monad.Fail.fail</code> as <code>Prelude.fail</code> and <code>Control.Monad.fail</code></li><br />
<li><code>Control.Monad.Fail</code> is now a redundant module that can be considered deprecated.</li><br />
</ul><br />
</li><br />
</ol><br />
<br />
=== Current status ===<br />
<br />
* [https://wiki.haskell.org/ZuriHac2015 ZuriHac 2015 (29.5. - 31.5.)]: Franz Thoma (@fmthoma) and me (David Luposchainsky aka @quchen) started implementing the MFP in GHC.<br />
** Desugaring to the new <code>fail</code> can be controlled via a new language extension, <code>MonadFailDesugaring</code><br />
** If the language extension is turned off, a warning will be emitted for code that would break if it was enabled.<br />
** Warnings are emitted for types that ''have'' a ''MonadFail'' instance. This still needs to be fixed.<br />
** The error messages are readable, but should be more so. We're still on this.<br />
* 2015-06-09: Estimated breakage by compiling Stackage. Smaller than expected.<br />
* 2015-06-09 (late): Published. People seem to like the idea, with a couple of pain points remaining.<br />
* 2015-06-16: [https://github.com/quchen/articles/blob/master/monad_fail_update1.md Update 1 posted.]<br />
* 2015-09-18: [https://phabricator.haskell.org/D1248 Patch nearly finished. Some nontrivial tests still fail.]</div>Blackouthttps://wiki.haskell.org/index.php?title=ZuriHac2015/Attendees&diff=59532ZuriHac2015/Attendees2015-03-14T19:50:47Z<p>Blackout: Add affiliation for David Luposchainsky (once again)</p>
<hr />
<div>{|<br />
|-<br />
! Name !! Affiliation<br />
|-<br />
| Gabor Greif || Alcatel-Lucent<br />
|-<br />
| Ian-Woo Kim || CERN<br />
|-<br />
| Alexander Kubelskyi || CERN<br />
|-<br />
| Daniel Lovasko || CERN<br />
|-<br />
| Jacopo Notarstefano || CERN<br />
|-<br />
| Blazej Kolad || CERN / Geneva Haskell Group <br />
|-<br />
| Pierre Radermecker || CIRB<br />
|-<br />
| Thomas Koch || comsolit AG, Debian<br />
|-<br />
| Jinhyuk Choi || ETH Zurich<br />
|-<br />
| Francesco Mazzoli || FPComplete<br />
|-<br />
| Jannis Limperg || Freiburg University<br />
|-<br />
| Oliver Charles || Fynder.io<br />
|-<br />
| Jean-Baptiste Giraudeau || Geneva Haskell Group<br />
|-<br />
| Cezary Bartoszuk || Google<br />
|-<br />
| Alexander Bernauer || Google<br />
|-<br />
| Lewin Bormann || Google<br />
|-<br />
| Clemens Fruhwirth || Google<br />
|-<br />
| Ruslan Khayrov || Google<br />
|-<br />
| Lennart Kolmodin || Google<br />
|-<br />
| Robin Palotai || Google<br />
|-<br />
| Daniel Pek || Google<br />
|-<br />
| Gleb Peregud || Google<br />
|-<br />
| Tiziano Santoro || Google<br />
|-<br />
| Andrew Shulaev || Google<br />
|-<br />
| Roland Szabo || Google<br />
|-<br />
| Michal Terepeta || Google<br />
|-<br />
| Johan Tibell || Google<br />
|-<br />
| Martijn van Steenbergen || Google UK<br />
|-<br />
| Jasper Van der Jeugt || Haskell Contractor<br />
|-<br />
| Marcus Hüsgen || Holidaycheck<br />
|-<br />
| Mark Korondi || IBM Research<br />
|-<br />
| Simon Meier || IBM Research<br />
|-<br />
| Andraž Bajt || Koofr<br />
|-<br />
| Miëtek Bak || Least Fixed<br />
|-<br />
| Julien Truffaut || Leonteq<br />
|-<br />
| Gerolf Seitz || Leonteq Securities AG <br />
|-<br />
| Johannes Drever || Linova Software<br />
|-<br />
| Bas van Dijk || LumiGuide<br />
|-<br />
| Edward Kmett || McGraw Hill Financial<br />
|-<br />
| Dario Bertini || NCCGroup<br />
|-<br />
| Mihaly Barasz || Nilcons<br />
|-<br />
| Gergely Risko || Nilcons<br />
|-<br />
| Leonard Punt || Q42<br />
|-<br />
| Sjoerd Visscher || Q42<br />
|-<br />
| Paul Körbitz || Qmedify GmbH<br />
|-<br />
| Christian Fischer || Software Engineer<br />
|-<br />
| Mario Pastorelli || Teralytics AG<br />
|-<br />
| Alexander Lehmann || TNG Technology Consulting<br />
|-<br />
| Franz Thoma || TNG Technology Consulting<br />
|-<br />
| David Luposchainsky || TNG Technology Consulting<br />
|-<br />
| Alexander Vershilov || Tweag I/O<br />
|-<br />
| Julien Cretel || University College Cork<br />
|-<br />
| Ivan Perez || University of Nottingham, Keera Studios<br />
|-<br />
| Jurgen Cito || University of Zurich<br />
|-<br />
| Amanj Sherwany || Università della Svizzera italiana (University of Lugano -USI)<br />
|-<br />
| Duncan Coutts || Well-Typed<br />
|-<br />
| Andres Löh || Well-Typed LLP<br />
|-<br />
| Michal Kawalec || X-Team<br />
|-<br />
| Yuriy Syrovetskiy || Yandex<br />
|-<br />
| Julian K. Arni || Zalora<br />
|-<br />
| Mathieu Boespflug || <br />
|-<br />
| Dino Burger || <br />
|-<br />
| Jaroslaw Buryk || <br />
|-<br />
| Tomas Carnecky || <br />
|-<br />
| Alois Cochard || <br />
|-<br />
| Péter Diviánszky || <br />
|-<br />
| Kaspar Emanuel || <br />
|-<br />
| Urska Erklavec || <br />
|-<br />
| Matthias Fischmann || <br />
|-<br />
| Martin Geisler || <br />
|-<br />
| Niklas Hambüchen || <br />
|-<br />
| Csaba Hruska || <br />
|-<br />
| Heinrich Hördegen || <br />
|-<br />
| Götz Isenmann || <br />
|-<br />
| Moritz Kiefer || <br />
|-<br />
| Vijay Kiran || <br />
|-<br />
| Alexander Kosenkov || <br />
|-<br />
| Bartosz Krol || <br />
|-<br />
| Michel Kuhlmann || <br />
|-<br />
| Mateusz Lenik || <br />
|-<br />
| Markus Liedl ||<br />
|-<br />
| Jeronimo Martinez || <br />
|-<br />
| Stanley Mwangi || <br />
|-<br />
| Jussi Mäki || <br />
|-<br />
| Carlo Nucera || <br />
|-<br />
| Filippo Perugini || <br />
|-<br />
| Romain Petit || <br />
|-<br />
| Tobias Pflug || <br />
|-<br />
| Blaz Repas || <br />
|-<br />
| Roland Senn || <br />
|-<br />
| Ganesh Sittampalam || <br />
|-<br />
| Luite Stegeman || <br />
|-<br />
| Dominic Steinitz || <br />
|-<br />
| Harald Steinlechner || <br />
|-<br />
| Luke Taylor || <br />
|-<br />
| Maarten Veenstra || <br />
|-<br />
| Tomas Vestelind || <br />
|-<br />
| Jasper Woudenberg || <br />
|-<br />
| Ilya Yanok || <br />
|-<br />
| Jakub Zalewski || <br />
|-<br />
| Robert Zaremba || <br />
|}</div>Blackouthttps://wiki.haskell.org/index.php?title=ZuriHac2015/Attendees&diff=59506ZuriHac2015/Attendees2015-03-07T17:44:28Z<p>Blackout: Add affiliation for David Luposchainsky</p>
<hr />
<div>{|<br />
|-<br />
! Name !! Affiliation<br />
|-<br />
| Gabor Greif || Alcatel-Lucent<br />
|-<br />
| Ian-Woo Kim || CERN<br />
|-<br />
| Alexander Kubelskyi || CERN<br />
|-<br />
| Daniel Lovasko || CERN<br />
|-<br />
| Jacopo Notarstefano || CERN<br />
|-<br />
| Blazej Kolad || CERN / Geneva Haskell Group <br />
|-<br />
| Pierre Radermecker || CIRB<br />
|-<br />
| Thomas Koch || comsolit AG, Debian<br />
|-<br />
| Francesco Mazzoli || FPComplete<br />
|-<br />
| Jannis Limperg || Freiburg University<br />
|-<br />
| Oliver Charles || Fynder.io<br />
|-<br />
| Jean-Baptiste Giraudeau || Geneva Haskell Group<br />
|-<br />
| Cezary Bartoszuk || Google<br />
|-<br />
| Alexander Bernauer || Google<br />
|-<br />
| Lewin Bormann || Google<br />
|-<br />
| Clemens Fruhwirth || Google<br />
|-<br />
| Ruslan Khayrov || Google<br />
|-<br />
| Lennart Kolmodin || Google<br />
|-<br />
| Robin Palotai || Google<br />
|-<br />
| Daniel Pek || Google<br />
|-<br />
| Gleb Peregud || Google<br />
|-<br />
| Tiziano Santoro || Google<br />
|-<br />
| Andrew Shulaev || Google<br />
|-<br />
| Michal Terepeta || Google<br />
|-<br />
| Johan Tibell || Google<br />
|-<br />
| Martijn van Steenbergen || Google UK<br />
|-<br />
| Jasper Van der Jeugt || Haskell Contractor<br />
|-<br />
| Marcus Hüsgen || Holidaycheck<br />
|-<br />
| Mark Korondi || IBM Research<br />
|-<br />
| Simon Meier || IBM Research<br />
|-<br />
| Andraž Bajt || Koofr<br />
|-<br />
| Miëtek Bak || Least Fixed<br />
|-<br />
| Julien Truffaut || Leonteq<br />
|-<br />
| Gerolf Seitz || Leonteq Securities AG <br />
|-<br />
| Bas van Dijk || LumiGuide<br />
|-<br />
| Edward Kmett || McGraw Hill Financial<br />
|-<br />
| Dario Bertini || NCCGroup<br />
|-<br />
| Mihaly Barasz || Nilcons<br />
|-<br />
| Gergely Risko || Nilcons<br />
|-<br />
| Leonard Punt || Q42<br />
|-<br />
| Sjoerd Visscher || Q42<br />
|-<br />
| Paul Körbitz || Qmedify GmbH<br />
|-<br />
| Christian Fischer || Software Engineer<br />
|-<br />
| Mario Pastorelli || Teralytics AG<br />
|-<br />
| Alexander Lehmann || TNG Technology Consulting<br />
|-<br />
| David Luposchainsky || TNG Technology Consulting<br />
|-<br />
| Alexander Vershilov || Tweag I/O<br />
|-<br />
| Julien Cretel || University College Cork<br />
|-<br />
| Ivan Perez || University of Nottingham, Keera Studios<br />
|-<br />
| Jurgen Cito || University of Zurich<br />
|-<br />
| Amanj Sherwany || Università della Svizzera italiana (University of Lugano -USI)<br />
|-<br />
| Duncan Coutts || Well-Typed<br />
|-<br />
| Michal Kawalec || X-Team<br />
|-<br />
| Yuriy Syrovetskiy || Yandex<br />
|-<br />
| Julian K. Arni || Zalora<br />
|-<br />
| Mathieu Boespflug || <br />
|-<br />
| Dino Burger || <br />
|-<br />
| Jaroslaw Buryk || <br />
|-<br />
| Tomas Carnecky || <br />
|-<br />
| Alois Cochard || <br />
|-<br />
| Péter Diviánszky || <br />
|-<br />
| Kaspar Emanuel || <br />
|-<br />
| Urska Erklavec || <br />
|-<br />
| Martin Geisler || <br />
|-<br />
| Niklas Hambüchen || <br />
|-<br />
| Csaba Hruska || <br />
|-<br />
| Götz Isenmann || <br />
|-<br />
| Moritz Kiefer || <br />
|-<br />
| Vijay Kiran || <br />
|-<br />
| Alexander Kosenkov || <br />
|-<br />
| Bartosz Krol || <br />
|-<br />
| Michel Kuhlmann || <br />
|-<br />
| Mateusz Lenik ||<br />
|-<br />
| Jeronimo Martinez || <br />
|-<br />
| Stanley Mwangi || <br />
|-<br />
| Jussi Mäki || <br />
|-<br />
| Carlo Nucera || <br />
|-<br />
| Romain Petit || <br />
|-<br />
| Tobias Pflug || <br />
|-<br />
| Blaz Repas || <br />
|-<br />
| Roland Senn || <br />
|-<br />
| Ganesh Sittampalam || <br />
|-<br />
| Luite Stegeman || <br />
|-<br />
| Dominic Steinitz || <br />
|-<br />
| Luke Taylor || <br />
|-<br />
| Maarten Veenstra || <br />
|-<br />
| Jasper Woudenberg || <br />
|-<br />
| Ilya Yanok || <br />
|-<br />
| Jakub Zalewski || <br />
|-<br />
| Robert Zaremba || <br />
|}</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=59345Functor-Applicative-Monad Proposal2015-02-16T21:21:37Z<p>Blackout: AMP successfully implemented in 7.10</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stood out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article describes the Haskell 2014 Applicative => Monad proposal (AMP)''', which was then implemented in GHC 7.10.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <s><hask>join</hask> is promoted into the Monad typeclass.</s> - Left out due to nontrivial breaking interaction with [[Roles]]. Will "probably be fixed in the future", but for now it's off the table.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== Future-proofing current code ==<br />
<br />
GHC 7.8 will issue two types of warnings in order to encourage wide-scale code fixing. The following describes how to get rid of them, and as a result ensures your code builds both now and after the AMP is finished.<br />
<br />
=== Missing superclasses ===<br />
<br />
(Warnings of the type "Warning: X is an instance of C, but not D")<br />
<br />
* Add Applicative/Functor instances for all your Monads. You can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
import Control.Applicative (Applicative(..))<br />
import Control.Monad (liftM, ap)<br />
<br />
-- Monad m<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadPlus. This can again be done easily using<br />
<haskell><br />
import Control.Applicative (Alternative(..))<br />
import Control.Monad (mzero, mplus)<br />
<br />
-- MonadPlus m<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
=== Future Prelude names ===<br />
<br />
"The name X clashes with a future Prelude name" - Prelude will export functions called <hask><*></hask>, <hask>join</hask> and <hask>pure</hask>, so if a module defines its own versions of them, there will be name clashes. There are multiple ways of getting rid of this type of warning (in a future-proof way).<br />
<br />
# Change your code to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
# Import Prelude definitions you need explicitly. For example, <hask>import Prelude (map, (+))</hask> would not import <hask>join</hask>, so no warning is issued as the module is compatible with the Prelude exporting <hask>join</hask>. <hask>hiding</hask>.<br />
# Due to GHC internals, you cannot use <hask>hiding (join, (<*>), pure)</hask> to silence the warnings, although this method would be future-proof. If you want to use <hask>hiding</hask>, you will have to silence the warnings using a sledgehammer <code>-fno-warn-amp</code> compiler flag. (If you do so make sure you know what you're doing, otherwise your module ''will'' break in 7.10.) To make 7.10 not complain about the then unrecognized flag, it's best to specify it in a CPP block,<br />
<haskell><br />
{-# LANGUAGE CPP #-}<br />
#if __GLASGOW_HASKELL__ >= 707 && __GLASGOW_HASKELL__ < 710<br />
{-# OPTIONS_GHC -fno-warn-amp #-}<br />
#endif<br />
</haskell><br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requires <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - ''sigh''<br />
<br />
With the new hierarchy, the answer would ''always'' be "use the least restrictive one".<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Num_instance_for_functions&diff=59166Num instance for functions2014-12-28T01:24:26Z<p>Blackout: Refurbish very patchy article to contain the essentials</p>
<hr />
<div>It is possible to write a <hask>Num</hask> instance for functions in Haskell:<br />
<br />
<haskell><br />
instance Num b => Num (a -> b) where<br />
negate = fmap negate<br />
(+) = liftA2 (+)<br />
(*) = liftA2 (*)<br />
fromInteger = pure . fromInteger<br />
abs = fmap abs<br />
signum = fmap signum<br />
</haskell><br />
<br />
This allows writing expressions of the form<br />
<br />
<haskell><br />
> (sin^2 + cos^2) 123.4<br />
1.0<br />
</haskell><br />
<br />
where functions can be added like <hask>Num</hask>bers, leaving their arguments open for later.<br />
<br />
The usefulness of this instance is debatable, and it is not currently part of [[Language and library specification|the language report]] or defined in any popular library.<br />
<br />
== See also ==<br />
<br />
* The [[applicative-numbers]] package, which generates numeric class instances for arbitrary [[applicative functor]]s (including functions).<br />
* http://www.haskell.org/pipermail/haskell-cafe/2006-November/019374.html<br />
* http://www.haskell.org/pipermail/haskell-cafe/2006-October/019105.html<br />
* http://www.haskell.org/pipermail/haskell-cafe/2001-February/001531.html<br />
* http://augustss.blogspot.com/2009/02/regression-they-say-that-as-you-get.html<br />
<br />
[[Category:Humor]]<br />
[[Category:FAQ]]<br />
[[Category:Style]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Programming_guidelines&diff=58588Programming guidelines2014-07-16T23:16:09Z<p>Blackout: "Multiple definitions" style is widely used, saying "prefer explicit case" is not right and no good reason to do so is given. Away with that misleading section.</p>
<hr />
<div>Programming guidelines shall help to make the code of a project better<br />
readable and maintainable by the varying number of contributors.<br />
<br />
It takes some programming experience to develop something like a<br />
personal "coding style" and guidelines only serve as rough shape for<br />
code. Guidelines should be followed by all members working on the<br />
project even if they prefer (or are already used to) different<br />
guidelines.<br />
<br />
These guidelines have been originally set up for the hets-project<br />
[http://www.informatik.uni-bremen.de/cofi/hets/ hets-project] and are<br />
now put on the [http://haskell.org/haskellwiki/ HaskellWiki] gradually<br />
integrating parts of the old hawiki<br />
entries [http://haskell.org/haskellwiki/Things_to_avoid ThingsToAvoid] and<br />
HaskellStyle (hopefully not<br />
hurting someone's copyrights). The other related entry<br />
TipsAndTricks treats more<br />
specific points that are left out here,<br />
<br />
Surely some style choices are a bit arbitrary (or "religious") and<br />
too restrictive with respect to language extensions. Nevertheless I hope<br />
to keep up these guidelines (at least as a basis) for our project<br />
in order to avoid maintaining diverging guidelines. Of course I want<br />
to supply - partly tool-dependent - reasons for certain decisions and<br />
also show alternatives by possibly bad examples. At the time of<br />
writing I use ghc-6.4.1, haddock-0.7 and (GNU-) emacs with the latest<br />
[http://www.haskell.org/haskell-mode/ haskell mode].<br />
<br />
The following quote and links are taken from<br />
HaskellStyle:<br />
<br />
We all have our own ideas about good Haskell style. There's More Than<br />
One Way To Do It. But some ways are better than others.<br />
<br />
Some comments from the GHC team about their internal coding<br />
standards can be found at<br />
http://hackage.haskell.org/trac/ghc/wiki/WorkingConventions<br />
<br />
Also http://research.microsoft.com/~simonpj/papers/haskell-retrospective/<br />
contains some brief comments on syntax and style,<br />
<br />
What now follows are descriptions of program documentation, file<br />
format, naming conventions and good programming practice (adapted form<br />
Matt's C/C++ Programming Guidelines and the Linux kernel coding<br />
style).<br />
<br />
<br />
=== Documentation ===<br />
<br />
<br />
Comments are to be written in application terms (i.e. user's point of<br />
view). Don't use technical terms - that's what the code is for!<br />
<br />
Comments should be written using correct spelling and grammar in complete<br />
sentences with punctation (in English only).<br />
<br />
"Generally, you want your comments to tell WHAT your code does, not HOW.<br />
Also, try to avoid putting comments inside a function body: if the<br />
function is so complex that you need to separately comment parts of it,<br />
you should probably" (... decompose it)<br />
<br />
Put a haddock comment on top of every exported function and data type!<br />
Make sure haddock accepts these comments.<br />
<br />
<br />
=== File Format ===<br />
<br />
<br />
All Haskell source files start with a haddock header of the form:<br />
<br />
<pre><br />
{- |<br />
Module : <File name or $Header$ to be replaced automatically><br />
Description : <optional short text displayed on contents page><br />
Copyright : (c) <Authors or Affiliations><br />
License : <license><br />
<br />
Maintainer : <email><br />
Stability : unstable | experimental | provisional | stable | frozen<br />
Portability : portable | non-portable (<reason>)<br />
<br />
<module description starting at first column><br />
-}<br />
</pre><br />
<br />
A possible compiler pragma (like {-# LANGUAGE CPP #-}) may precede<br />
this header. The following hierarchical module name must of course<br />
match the file name.<br />
<br />
Make sure that the description is changed to meet the module (if the<br />
header was copied from elsewhere). Insert your email address as maintainer.<br />
<br />
Try to write portable (Haskell98) code. If you use e.g. multi-parameter<br />
type classes and functional dependencies the code becomes<br />
"non-portable (MPTC with FD)".<br />
<br />
The \$Header\$ entry will be automatically expanded.<br />
<br />
Lines should not be longer than 80 (preferably 75)<br />
characters to avoid wrapped lines (for casual readers)!<br />
<br />
Don't leave trailing white space in your code in every line.<br />
<br />
Expand all your tabs to spaces to avoid the danger of wrongly expanding<br />
them (or a different display of tabs versus eight spaces). Possibly put<br />
something like the following in your ~/.emacs file.<br />
<br />
(custom-set-variables '(indent-tabs-mode nil))<br />
<br />
The last character in your file should be a newline! Under solaris<br />
you'll get a warning if this is not the case and sometimes last lines<br />
without newlines are ignored (i.e. "#endif" without newline). Emacs<br />
usually asks for a final newline.<br />
<br />
You may use http://hackage.haskell.org/package/scan to check your file format.<br />
<br />
The whole module should not be too long (about 400 lines)<br />
<br />
=== Naming Conventions ===<br />
<br />
<br />
In Haskell types start with capital and functions with lowercase<br />
letters, so only avoid infix identifiers! Defining symbolic infix<br />
identifiers should be left to library writers only.<br />
<br />
(The infix identifier "\\" at the end of a line causes cpp preprocessor<br />
problems.)<br />
<br />
Names (especially global ones) should be descriptive and if you need<br />
long names write them as mixed case words (aka camelCase). (but "tmp"<br />
is to be preferred over "thisVariableIsATemporaryCounter")<br />
<br />
Also in the standard libraries, function names with multiple words are<br />
written using the camelCase convention. Similarly, type, typeclass and<br />
constructor names are written using the StudlyCaps convention.<br />
<br />
Some parts of our code use underlines (without unnecessary uppercase<br />
letters) for long identifiers to better reflect names given with<br />
hyphens in the requirement documentation. Also such names should be<br />
transliterated to camlCase identifiers possibly adding a (consistent)<br />
suffix or prefix to avoid conflicts with keywords. However, instead of<br />
a recurring prefix or suffix you may consider to use qualified imports<br />
and names.<br />
<br />
<br />
=== Good Programming Practice ===<br />
<br />
<br />
"Functions should be short and sweet, and do just one thing. They should<br />
fit on one or two screenfuls of text (the ISO/ANSI screen size is 80x24,<br />
as we all know), and do one thing and do that well."<br />
<br />
Most haskell functions should be at most a few lines, only case<br />
expression over large data types (that should be avoided, too) may need<br />
corresponding space.<br />
<br />
The code should be succinct (though not obfuscated), readable and easy to<br />
maintain (after unforeseeable changes). Don't exploit exotic language<br />
features without good reason.<br />
<br />
It's not fixed how deep you indent (4 or 8 chars). You can break the<br />
line after "do", "let", "where", and "case .. of". Make sure that<br />
renamings don't destroy your layout. (If you get to far to the right,<br />
the code is unreadable anyway and needs to be decomposed.)<br />
<br />
Bad:<br />
case foo of Foo -> "Foo"<br />
Bar -> "Bar"<br />
Good:<br />
case <longer expression> of<br />
Foo -> "Foo"<br />
Bar -> "Bar"<br />
<br />
Avoid the notation with braces and semicolons since the layout rule<br />
forces you to properly align your alternatives.<br />
<br />
Respect compiler warnings. Supply type signatures, avoid shadowing and<br />
unused variables. Particularly avoid non-exhaustive and<br />
overlapping patterns. Missing unreachable cases can be filled in using<br />
"error" with a fixed string "<ModuleName>.<function>" to indicate the<br />
error position (in case the impossible should happen). Don't invest<br />
time to "show" the offending value, only do this temporarily when<br />
debugging the code.<br />
<br />
Don't leave unused or commented-out code in your files! Readers don't<br />
know what to think of it.<br />
<br />
<br />
<br />
==== Partial functions ====<br />
<br />
For partial functions do document their preconditions (if not obvious)<br />
and make sure that partial functions are only called when<br />
preconditions are obviously fulfilled (i.e. by a case statement or a<br />
previous test). Particularly the call of "head" should be used with<br />
care or (even better) be made obsolete by a case statement.<br />
<br />
Usually a case statement (and the import of isJust and fromJust from<br />
Data.[[Maybe]]) can be avoided by using the "maybe" function:<br />
<br />
maybe (error "<ModuleName>.<function>") id $ Map.lookup key map<br />
<br />
Generally we require you to be more explicit about failure<br />
cases. Surely a missing (or an irrefutable) pattern would precisely<br />
report the position of a runtime error, but these are not so obvious<br />
when reading the code.<br />
<br />
==== Let or where expressions ====<br />
<br />
Do avoid mixing and nesting "let" and "where". (I prefer the<br />
expression-stylistic "let".) Use auxiliary top-level functions that<br />
you do not export. Export lists also support the detection of unused<br />
functions.<br />
<br />
<br />
==== Code reuse ====<br />
<br />
If you notice that you're doing the same task again, try to generalize<br />
it in order to avoid duplicate code. It is frustrating to change the<br />
same error in several places.<br />
<br />
<br />
==== Application notation ====<br />
<br />
Many parentheses can be eliminated using the infix application operator "$"<br />
with lowest priority. Try at least to avoid unnecessary parentheses in<br />
standard infix expression.<br />
<br />
f x : g x ++ h x<br />
<br />
a == 1 && b == 1 || a == 0 && b == 0<br />
<br />
Rather than putting a large final argument in parentheses (with a<br />
distant closing one) consider using "$" instead.<br />
<br />
"f (g x)" becomes "f $ g x" and consecutive applications<br />
"f (g (h x))" can be written as "f $ g $ h x" or "f . g $ h x".<br />
<br />
A function definition like<br />
"f x = g $ h x" can be abbreviated to "f = g . h".<br />
<br />
Note that the final argument may even be an infix- or case expression:<br />
<br />
map id $ c : l<br />
<br />
filter (const True) . map id $ case l of ...<br />
<br />
However, be aware that $-terms cannot be composed further in infix<br />
expressions.<br />
<br />
Probably wrong:<br />
f $ x ++ g $ x<br />
<br />
But the scope of an expression is also limited by the layout rule, so<br />
it is usually safe to use "$" on right hand sides.<br />
<br />
Ok:<br />
do f $ l<br />
++<br />
do g $ l<br />
<br />
Of course "$" can not be used in types. GHC has also some primitive<br />
functions involving the kind "#" that cannot be applied using "$".<br />
<br />
Last warning: always leave spaces around "$" (and other mixfix<br />
operators) since a clash with template haskell is possible.<br />
<br />
(Also write "\ t" instead of "\t" in lambda expressions)<br />
<br />
<br />
==== List Comprehensions ====<br />
<br />
Use these only when "short and sweet". Prefer map, filter, and foldr!<br />
<br />
Instead of:<br />
<br />
[toUpper c | c <- s]<br />
<br />
write:<br />
<br />
map toUpper s<br />
<br />
<br />
Consider:<br />
<br />
[toUpper c | s <- strings, c <- s]<br />
<br />
Here it takes some time for the reader to find out which value depends<br />
on what other value and it is not so clear how many times the interim<br />
values s and c are used. In contrast to that the following can't be clearer:<br />
<br />
map toUpper (concat strings)<br />
<br />
<br />
When using higher order functions you can switch easier to data<br />
structures different from list. Compare:<br />
<br />
map (1+) list<br />
<br />
and:<br />
<br />
Set.map (1+) set<br />
<br />
<br />
==== Types ====<br />
<br />
Prefer proper data types over type synonyms or tuples even if you have<br />
to do more constructing and unpacking. This will make it easier to<br />
supply class instances later on. Don't put class constraints on<br />
a data type, constraints belong only to the functions that manipulate<br />
the data.<br />
<br />
Using type synonyms consistently is difficult over a longer time,<br />
because this is not checked by the compiler. (The types shown by<br />
the compiler may be unpredictable: i.e. FilePath, String or [Char])<br />
<br />
Take care if your data type has many variants (unless it is an<br />
enumeration type.) Don't repeat common parts in every variant since<br />
this will cause code duplication.<br />
<br />
Bad (to handle arguments in sync):<br />
<br />
data Mode f p = Box f p | Diamond f p<br />
<br />
Good (to handle arguments only once):<br />
<br />
data BoxOrDiamond = Box | Diamond<br />
<br />
data Mode f p = Mode BoxOrDiamond f p<br />
<br />
<br />
Consider (bad):<br />
<br />
data Tuple a b = Tuple a b | Undefined<br />
<br />
versus (better):<br />
<br />
data Tuple a b = Tuple a b<br />
<br />
and using:<br />
<br />
Maybe (Tuple a b)<br />
<br />
(or another monad) whenever an undefined result needs to be propagated<br />
<br />
<br />
==== Records ====<br />
<br />
For (large) records avoid the use of the constructor directly and<br />
remember that the order and number of fields may change.<br />
<br />
Take care with (the rare case of) depend polymorphic fields:<br />
<br />
data Fields a = VariantWithTwo<br />
{ field1 :: a<br />
, field2 :: a }<br />
<br />
The type of a value v can not be changed by only setting field1:<br />
<br />
v { field1 = f }<br />
<br />
Better construct a new value:<br />
<br />
VariantWithTwo { field1 = f } -- leaving field2 undefined<br />
<br />
Or use a polymorphic element that is instantiated by updating:<br />
<br />
empty = VariantWithTwo { field1 = [], field2 = [] }<br />
<br />
empty { field1 = [f] }<br />
<br />
Several variants with identical fields may avoid some code duplication<br />
when selecting and updating, though possibly not in a few<br />
depended polymorphic cases.<br />
<br />
However, I doubt if the following is a really good alternative to the<br />
above data Mode with data BoxOrDiamond.<br />
<br />
<br />
data Mode f p =<br />
Box { formula :: f, positions :: p }<br />
| Diamond { formula :: f, positions :: p }<br />
<br />
<br />
==== IO ====<br />
<br />
Try to strictly separate IO, Monad and pure (without do) function<br />
programming (possibly via separate modules).<br />
<br />
Bad:<br />
x <- return y<br />
...<br />
<br />
Good:<br />
let x = y<br />
...<br />
<br />
<br />
Don't use Prelude.interact and make sure your program does not depend<br />
on the (not always obvious) order of evaluation. E.g. don't read and<br />
write to the same file:<br />
<br />
This will fail:<br />
<br />
do s <- readFile f<br />
writeFile f $ 'a' : s<br />
<br />
because of lazy IO! (Writing is starting before reading is finished.)<br />
<br />
==== Trace ====<br />
<br />
Tracing is for debugging purposes only and should not be used as<br />
feedback for the user. Clean code is not cluttered by trace calls.<br />
<br />
<br />
==== Imports ====<br />
<br />
Standard library modules like Char. List, Maybe, Monad, etc should be<br />
imported by their hierarchical module name, i.e. the base package (so<br />
that haddock finds them):<br />
<br />
import Data.List<br />
import Control.Monad<br />
import System.Environment<br />
<br />
The libraries for Set and Map are to be imported qualified:<br />
<br />
import qualified Data.Set as Set<br />
import qualified Data.Map as Map<br />
<br />
<br />
==== Glasgow extensions and Classes ====<br />
<br />
[[Use of language extensions|Stay away from extensions]] as long as possible. Also use classes with<br />
care because soon the desire for overlapping instances (like for lists<br />
and strings) may arise. Then you may want MPTC (multi-parameter type<br />
classes), functional dependencies (FD), undecidable and possibly incoherent<br />
instances and then you are "in the wild" (according to SPJ).<br />
<br />
=== Style in other languages ===<br />
<br />
* [http://www.cs.caltech.edu/~cs20/a/style.html OCaml style]<br />
<br />
=== Final remarks ===<br />
<br />
Despite guidelines, writing "correct code" (without formal proof<br />
support yet) still remains the major challenge. As motivation to<br />
follow these guidelines consider the points that are from the "C++<br />
Coding Standard", where I replaced "C++" with "Haskell".<br />
<br />
Good Points:<br />
<br />
* programmers can go into any code and figure out what's going on<br />
<br />
* new people can get up to speed quickly<br />
<br />
* people new to Haskell are spared the need to develop a personal style and defend it to the death<br />
<br />
* people new to Haskell are spared making the same mistakes over and over again<br />
<br />
* people make fewer mistakes in consistent environments<br />
<br />
* programmers have a common enemy :-)<br />
<br />
Bad Points:<br />
<br />
* the standard is usually stupid because it was made by someone who doesn't understand Haskell<br />
<br />
* the standard is usually stupid because it's not what I do<br />
<br />
* standards reduce creativity<br />
<br />
* standards are unnecessary as long as people are consistent<br />
<br />
* standards enforce too much structure<br />
<br />
* people ignore standards anyway<br />
<br />
<br />
<br />
[[Category:Style]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=58553Functor-Applicative-Monad Proposal2014-07-08T21:47:14Z<p>Blackout: Join will not be part of Monad for now, strike element off the list</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <s><hask>join</hask> is promoted into the Monad typeclass.</s> - Left out due to nontrivial breaking interaction with [[Roles]]. Will "probably be fixed in the future", but for now it's off the table.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== Future-proofing current code ==<br />
<br />
GHC 7.8 will issue two types of warnings in order to encourage wide-scale code fixing. The following describes how to get rid of them, and as a result ensures your code builds both now and after the AMP is finished.<br />
<br />
=== Missing superclasses ===<br />
<br />
(Warnings of the type "Warning: X is an instance of C, but not D")<br />
<br />
* Add Applicative/Functor instances for all your Monads. You can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
import Control.Applicative (Applicative(..))<br />
import Control.Monad (liftM, ap)<br />
<br />
-- Monad m<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadPlus. This can again be done easily using<br />
<haskell><br />
import Control.Applicative (Alternative(..))<br />
import Control.Monad (mzero, mplus)<br />
<br />
-- MonadPlus m<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
=== Future Prelude names ===<br />
<br />
"The name X clashes with a future Prelude name" - Prelude will export functions called <hask><*></hask>, <hask>join</hask> and <hask>pure</hask>, so if a module defines its own versions of them, there will be name clashes. There are multiple ways of getting rid of this type of warning (in a future-proof way).<br />
<br />
# Change your code to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
# Import Prelude definitions you need explicitly. For example, <hask>import Prelude (map, (+))</hask> would not import <hask>join</hask>, so no warning is issued as the module is compatible with the Prelude exporting <hask>join</hask>. <hask>hiding</hask>.<br />
# Due to GHC internals, you cannot use <hask>hiding (join, (<*>), pure)</hask> to silence the warnings, although this method would be future-proof. If you want to use <hask>hiding</hask>, you will have to silence the warnings using a sledgehammer <code>-fno-warn-amp</code> compiler flag. (If you do so make sure you know what you're doing, otherwise your module ''will'' break in 7.10.) To make 7.10 not complain about the then unrecognized flag, it's best to specify it in a CPP block,<br />
<haskell><br />
{-# LANGUAGE CPP #-}<br />
#if __GLASGOW_HASKELL__ >= 707 && __GLASGOW_HASKELL__ < 710<br />
{-# OPTIONS_GHC -fno-warn-amp #-}<br />
#endif<br />
</haskell><br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - ''sigh''<br />
<br />
With the new hierarchy, the answer would ''always'' be "use the least restrictive one".<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== '''Done''': Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== '''Current stage''': Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=57884Functor-Applicative-Monad Proposal2014-04-16T20:53:38Z<p>Blackout: Update "current stage" headline, since GHC 7.8 implements AMP warnings</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== Future-proofing current code ==<br />
<br />
GHC 7.8 will issue two types of warnings in order to encourage wide-scale code fixing. The following describes how to get rid of them, and as a result ensures your code builds both now and after the AMP is finished.<br />
<br />
=== Missing superclasses ===<br />
<br />
(Warnings of the type "Warning: X is an instance of C, but not D")<br />
<br />
* Add Applicative/Functor instances for all your Monads. You can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
import Control.Applicative (Applicative(..))<br />
import Control.Monad (liftM, ap)<br />
<br />
-- Monad m<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadPlus. This can again be done easily using<br />
<haskell><br />
import Control.Applicative (Alternative(..))<br />
import Control.Monad (mzero, mplus)<br />
<br />
-- MonadPlus m<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
=== Future Prelude names ===<br />
<br />
"The name X clashes with a future Prelude name" - Prelude will export functions called <hask><*></hask>, <hask>join</hask> and <hask>pure</hask>, so if a module defines its own versions of them, there will be name clashes. There are multiple ways of getting rid of this type of warning (in a future-proof way).<br />
<br />
# Change your code to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
# Import Prelude definitions you need explicitly. For example, <hask>import Prelude (map, (+))</hask> would not import <hask>join</hask>, so no warning is issued as the module is compatible with the Prelude exporting <hask>join</hask>. <hask>hiding</hask>.<br />
# Due to GHC internals, you cannot use <hask>hiding (join, (<*>), pure)</hask> to silence the warnings, although this method would be future-proof. If you want to use <hask>hiding</hask>, you will have to silence the warnings using a sledgehammer <code>-fno-warn-amp</code> compiler flag. (If you do so make sure you know what you're doing, otherwise your module ''will'' break in 7.10.) To make 7.10 not complain about the then unrecognized flag, it's best to specify it in a CPP block,<br />
<haskell><br />
{-# LANGUAGE CPP #-}<br />
#if __GLASGOW_HASKELL__ >= 707 && __GLASGOW_HASKELL__ < 710<br />
{-# OPTIONS_GHC -fno-warn-amp #-}<br />
#endif<br />
</haskell><br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - ''sigh''<br />
<br />
With the new hierarchy, the answer would ''always'' be "use the least restrictive one".<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== '''Done''': Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== '''Current stage''': Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Debugging&diff=57736Debugging2014-04-02T13:37:28Z<p>Blackout: /* Stack trace */ More detailed description plus example</p>
<hr />
<div>== Stack trace ==<br />
<br />
<br />
<br />
=== General usage ===<br />
<br />
Recent versions of GHC allow a dump of a stack trace (of all cost centres) when an exception is raised. In order to enable this, compile with <code>-prof</code>, and run with <code>+RTS -xc</code>. (Since only the cost centre stack will be printed, you may want to add <code>-auto-all -caf-all</code> to the compilation step to include all definitions in the trace.)<br />
<br />
A more detailed list of options can be found [http://www.haskell.org/ghc/docs/latest/html/users_guide/runtime-control.html#rts-options-debugging in the RTS section] of the GHC user's guide.<br />
<br />
<br />
<br />
=== Example ===<br />
<br />
<haskell><br />
-- test.hs<br />
crash = sum [1,2,3,undefined,5,6,7,8]<br />
main = print crash<br />
</haskell><br />
<br />
<br />
> ghc-7.6.3 test.hs -prof -auto-all -caf-all && ./test +RTS -xc<br />
*** Exception (reporting due to +RTS -xc): (THUNK_2_0), stack trace: <br />
GHC.Err.CAF<br />
--> evaluated by: Main.crash,<br />
called from Main.CAF:crash_reH<br />
test: Prelude.undefined<br />
<br />
== Printf and friends ==<br />
<br />
The simplest approach is to use <tt>Debug.Trace.trace</tt>:<br />
<pre><br />
trace :: String -> a -> a<br />
</pre><br />
: "''When called, trace outputs the string in its first argument, before returning the second argument as its result.'"<br />
<br />
A common idiom to trace a function is:<br />
<pre><br />
myfun a b | trace ("myfun " ++ show a ++ " " ++ show b) False = undefined<br />
myfun a b = ...<br />
</pre><br />
The advantage is that disabling and enabling the trace takes only one line comment.<br />
<br />
You must keep in mind that due to lazy evaluation your traces will only print if the value they wrap is ever demanded.<br />
<br />
The trace function is located in the base package. The package [http://hackage.haskell.org/package/htrace htrace] defines a trace function similar to the one in the base package, but with indentation for better visual effect (see the [http://www.haskell.org/pipermail/haskell-cafe/2012-January/098847.html mailing list thread] for examples). Other tools can be found at [http://hackage.haskell.org/packages/archive/pkg-list.html#cat:debug the debug category in Hackage].<br />
<br />
A more powerful alternative for this approach is [http://hackage.haskell.org/package/hood Hood]. Even if it hasn't been<br />
updated in some time, Hood works perfectly with the current ghc<br />
distribution. Even more, Hugs has it already integrated, see the [http://cvs.haskell.org/Hugs/pages/users_guide/observe.html manual page]. Add an <tt>import Observe</tt> and start inserting observations in your code.<br />
For instance:<br />
<pre><br />
import Hugs.Observe<br />
<br />
f' = observe "Informative name for f" f <br />
f x = if odd x then x*2 else 0<br />
</pre><br />
And then in hugs:<br />
<pre><br />
Main> map f' [1..5]<br />
[2,0,6,0,10]<br />
<br />
>>>>>>> Observations <<<<<<<br />
<br />
Informative name for f<br />
{ \ 5 -> 10<br />
, \ 4 -> 0<br />
, \ 3 -> 6<br />
, \ 2 -> 0<br />
, \ 1 -> 2<br />
}<br />
<br />
</pre><br />
<br />
outputs a report of all the invocations of f and their result.<br />
<br />
I have a handy bogus Hugs.Observe module with no-ops for the observations so that I don't need to remove them manually, expecting that the compiler will optimize them away.<br />
<br />
The [http://hackage.haskell.org/package/GHood GHood] package adds a graphical back-end to Hood. See also the [http://community.haskell.org/~claus/GHood/ GHood homepage].<br />
<br />
<br />
== The Safe Library ==<br />
<br />
There is a safe library of functions from the Prelude that can crash, see [http://community.haskell.org/~ndm/safe/ the safe library]. If you get an error message such as "pattern match failure, head []", you can then use <tt>headNote "extra information"</tt> to get a more detailed error message for that particular call to <tt>head</tt>. The safe library also has functions that return default values and wrap their computation in <tt>Maybe</tt> as required.<br />
<br />
<br />
== Offline analysis of traces ==<br />
The most advanced debugging tools are based in offline analysis of traces. [http://projects.haskell.org/hat/ Hat] is probably the most up-to-date tool for this, offering a comprehensive set of tools. [[User:NeilMitchell|Neil Mitchell]] has made available a Windows port of Hat at [http://projects.haskell.org/~ndm/hat/ his site].<br />
<br />
The disadvantage of these tools is that they are not always compatible with the latest libraries, so you can put them to use only in some cases. <br />
<br />
''Some Hat user should complete this section''<br />
<br />
== Dynamic breakpoints in GHCi ==<br />
<br />
Finally, the [[GHC/GHCi debugger| GHCi debugger]] enables dynamic<br />
breakpoints and intermediate values observation. <br />
<br />
This tool allows to set breakpoints in your code, directly from the GHCi command prompt. An example session:<br />
<pre><br />
*main:Main> :break add Main 2<br />
Breakpoint set at (2,15)<br />
*main:Main> qsort [10,9..1]<br />
Local bindings in scope:<br />
x :: a, xs :: [a], left :: [a], right :: [a]<br />
<br />
qsort2.hs:2:15-46> :sprint x<br />
x = _<br />
qsort2.hs:2:15-46> x<br />
</pre><br />
This is an untyped, unevaluated computation. You can use <hask>seq</hask> to force its evaluation and then <code>:print</code> to recover its type<br />
<pre><br />
qsort2.hs:2:15-46> seq x ()<br />
() <br />
qsort2.hs:2:15-46> :p x<br />
x - 10<br />
</pre><br />
<br />
Once a breakpoint is hit, you can explore the bindings in scope, as well as to evaluate any Haskell expression, as you would do in a normal GHCi prompt. The <code>:print</code> command can be very useful to explore the laziness of your code.<br />
<br />
<br />
== Source-located errors ==<br />
<br />
[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/loch LocH] provides wrappers over<br />
<hask>assert</hask> for generating source-located exceptions and errors.<br />
<br />
Consider the use of a located <hask>fromJust</hask>:<br />
<br />
<haskell><br />
import Debug.Trace.Location<br />
import qualified Data.Map as M<br />
import Data.Maybe<br />
<br />
main = do print f<br />
<br />
f = let m = M.fromList<br />
[(1,"1")<br />
,(2,"2")<br />
,(3,"3")]<br />
s = M.lookup 4 m<br />
in fromJustSafe assert s<br />
<br />
fromJustSafe a s = check a (fromJust s)<br />
</haskell><br />
<br />
This will result in:<br />
<br />
<haskell><br />
$ ./a.out<br />
a.out: A.hs:12:20-25: Maybe.fromJust: Nothing<br />
</haskell><br />
<br />
This can be automated, using the 'loch' preprocessor, so a program<br />
failing with:<br />
<br />
<code><br />
$ ghc A.hs --make -no-recomp<br />
[1 of 1] Compiling Main ( A.hs, A.o )<br />
Linking A ...<br />
<br />
$ ./A<br />
A: Maybe.fromJust: Nothing<br />
</code><br />
<br />
Can be transformed to a src-located one by adding:<br />
<br />
<haskell><br />
import Debug.Trace.Location<br />
</haskell><br />
<br />
and then recompiling with the preprocessor on:<br />
<br />
<code><br />
$ ghc A.hs --make -pgmF loch -F -no-recomp<br />
[1 of 1] Compiling Main ( A.hs, A.o )<br />
Linking A ...<br />
<br />
$ ./A<br />
A: A.hs:14:14-19: Maybe.fromJust: Nothing<br />
</code><br />
<br />
<br />
== Other tricks ==<br />
<br />
* If you use GHC, you can get a stack trace in the console when your program fails with an error condition. See the [http://www.haskell.org/ghc/docs/latest/html/users_guide/runtime-control.html#rts-options-debugging description of relevant runtime options].<br />
* Some tips how to use GHCi debugger are also in [http://www.haskell.org/pipermail/glasgow-haskell-users/2009-February/016571.html this message].<br />
<br />
<br />
=== Locating a failure in a library function ===<br />
<br />
The simplest way to provide locating in the source code a mismatch<br />
run-time error in the library functions:<br />
<haskell><br />
head, tail, fromJust<br />
</haskell><br />
<br />
and others is to avoid these functions and to use explicit matching instead.<br />
<br />
For example, consider:<br />
<br />
<haskell><br />
g x = h $ fromJust $ f x,<br />
</haskell><br />
<br />
ghc-6.6 often loses the reference to <hask>g</hask>, <hask>f</hask>,<br />
and <hask>h</hask> in its run-time error report, when <hask>f</hask><br />
returns <hask>Nothing</hask>.<br />
<br />
But for the program:<br />
<br />
<haskell><br />
g x = let Just y = f x in h y,<br />
</haskell><br />
<br />
GHC reports:<br />
<br />
<haskell><br />
Main: M1.hs:9:11-22:<br />
Irrefutable pattern failed for pattern Data.Maybe.Just y<br />
</haskell><br />
<br />
Indicating the source of the failure.<br />
<br />
<br />
=== Mysterious parse errors ===<br />
<br />
GHC provides `-ferror-spans`, which will give you the exactly position<br />
of the start and end of an offending statement.<br />
<br />
<br />
=== Infinite loops ===<br />
On glasgow-haskell-users on 21 Nov 2007, pepe made the following suggestion for detecting the cause infinite loops in GHCi. Assuming the offending function is named `loop`, and takes one argument:<br />
<br />
# enable the flag -fbreak-on-error (`:set -fbreak-on-error` in GHCi)<br />
# run your expression with :trace (`:trace loop 'a'`)<br />
# hit Ctrl-C while your program is stuck in the loop to have the debugger break in the loop<br />
# use :history and :back to find out where the loop is located and why.<br />
<br />
''(For which versions? ghci >= 6.8?)''<br />
<br />
[[Category:Tools]]<br />
[[Category:Development tools]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Pronunciation&diff=57318Pronunciation2013-12-17T23:21:37Z<p>Blackout: Remove trailing whitespace in <hask> tags</p>
<hr />
<div>There was a thread on Haskell-Cafe about [http://www.haskell.org/pipermail/haskell-cafe/2008-January/038756.html how to pronounce Haskell].<br />
<br />
Below are some notes for beginners on how to pronounce those strange Haskell operators and 'read' Haskell programs.<br />
<br />
This is meant to be a table with formal and informal ways of saying various operators and code snippets.<br />
<br />
{|<br />
! Symbol<br />
! Pronunciation<br />
! References<br />
|-<br />
| <hask>::</hask><br />
| has type (in definitions); at type (in expressions or patterns)<br />
|-<br />
| <hask>-></hask><br />
| maps to, to<br />
|-<br />
| <hask>=</hask><br />
| is<br />
|-<br />
| <hask>==</hask><br />
| equals<br />
| [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#v%3A-61--61- Prelude]<br />
|-<br />
| <hask>/=</hask><br />
| not equals<br />
| [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#v%3A-61--61- Prelude]<br />
|-<br />
| <hask>=></hask><br />
| is a witness for, implies<br />
| [http://www.haskell.org/tutorial/classes.html Type Classes and Overloading]<br />
|-<br />
| <hask>.</hask><br />
| dot (could be used anywhere, but especially in, for example, Data.Char.ord), ring, compose (for example, negate . (+1)), (silent) (for example, forall a. (Num a) => a)<br />
|-<br />
| <hask><-</hask><br />
| drawn from, from<br />
|-<br />
| <hask>-<</hask><br />
| arrow application<br />
| [http://haskell.org/arrows/syntax.html Arrows syntax]<br />
|-<br />
| <hask>&&&</hask><br />
| both, fanout<br />
| [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html Control.Arrow]<br />
|-<br />
| <hask>|||</hask><br />
| either, fanin<br />
| [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html#v%3A%7C%7C%7C Control.Arrow]<br />
|<br />
|-<br />
| <hask>++</hask><br />
| append<br />
| [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#v%3A-43--43- Prelude]<br />
|-<br />
| <hask>>>=</hask><br />
| bind<br />
| [http://www.haskell.org/tutorial/monads.html About monads]<br />
[[Monads as containers]]<br />
|-<br />
| <hask>>></hask><br />
| then, sequence<br />
| [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#v:-62--62- Prelude]<br />
|-<br />
| <hask><*></hask><br />
| applied over<br />
| [http://www.haskell.org/haskellwiki/Applicative_functor Applicative Functors]<br />
|-<br />
| <hask>\</hask><br />
| lambda<br />
| [[Lambda abstraction]]<br />
[http://www.haskell.org/onlinereport/exps.html#sect3.3 Haskell 98 Report]<br />
|-<br />
| <hask>!</hask><br />
| bang; strict (in patterns or data definitions); index (in expressions)<br />
| [http://www.haskell.org/ghc/docs/latest/html/users_guide/bang-patterns.html Bang Patterns]<br />
[[Keywords]]<br />
|-<br />
| <hask>~</hask><br />
| irrefutable, lazy (in patterns)<br />
|-<br />
| <hask>:</hask><br />
| cons<br />
|-<br />
| <hask>[]</hask><br />
| nil<br />
|-<br />
| <hask>()</hask><br />
| unit<br />
|-<br />
| <hask>(,)</hask><br />
| 2-tuple, pair<br />
|-<br />
| <hask>(a,b,c)</hask><br />
| [3-]tuple [of] a, b, and c<br />
|-<br />
| <hask>({)}</hask><br />
| just as inconvenient to convey grouping verbally, whether it's layout or punctuation<br />
|}<br />
<br />
{|<br />
! Example<br />
! Pronunciation<br />
|-<br />
| <hask>f :: Int -> Int</hask><br />
| f has type Int to Int<br />
|}<br />
<br />
Thoughts on improving this page:<br />
<br />
The tables above would be best split into more columns to distinguish Informal, possibly bad suggestions like "then", "is", "gets", from Formal correct ways of saying the same thing. The Symbols could also be named in a literal way in another column, such as "arrow" or "double-colon". The Description column can be quite brief and a link provided to the relevant wiki page for the operator.<br />
<br />
Some words, particularly the informal ones, may be good for several different symbols, but these can hopefully be arranged so that their context will make them unambiguous when reading a code snippet.<br />
<br />
[[Category:Syntax]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=57165Functor-Applicative-Monad Proposal2013-12-03T13:32:16Z<p>Blackout: /* Beginner friendliness */ Markdown -> Wiki markup: *x* to ''x''</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== Future-proofing current code ==<br />
<br />
GHC 7.8 will issue two types of warnings in order to encourage wide-scale code fixing. The following describes how to get rid of them, and as a result ensures your code builds both now and after the AMP is finished.<br />
<br />
=== Missing superclasses ===<br />
<br />
(Warnings of the type "Warning: X is an instance of C, but not D")<br />
<br />
* Add Applicative/Functor instances for all your Monads. You can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
import Control.Applicative (Applicative(..))<br />
import Control.Monad (liftM, ap)<br />
<br />
-- Monad m<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadPlus. This can again be done easily using<br />
<haskell><br />
import Control.Applicative (Alternative(..))<br />
import Control.Monad (mzero, mplus)<br />
<br />
-- MonadPlus m<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
=== Future Prelude names ===<br />
<br />
"The name X clashes with a future Prelude name" - Prelude will export functions called <hask><*></hask>, <hask>join</hask> and <hask>pure</hask>, so if a module defines its own versions of them, there will be name clashes. There are multiple ways of getting rid of this type of warning (in a future-proof way).<br />
<br />
# Change your code to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
# Import Prelude definitions you need explicitly. For example, <hask>import Prelude (map, (+))</hask> would not import <hask>join</hask>, so no warning is issued as the module is compatible with the Prelude exporting <hask>join</hask>. <hask>hiding</hask>.<br />
# Due to GHC internals, you cannot use <hask>hiding (join, (<*>), pure)</hask> to silence the warnings, although this method would be future-proof. If you want to use <hask>hiding</hask>, you will have to silence the warnings using a sledgehammer <code>-fno-warn-amp</code> compiler flag. (If you do so make sure you know what you're doing, otherwise your module ''will'' break in 7.10.) To make 7.10 not complain about the then unrecognized flag, it's best to specify it in a CPP block,<br />
<haskell><br />
{-# LANGUAGE CPP #-}<br />
#if __GLASGOW_HASKELL__ >= 707 && __GLASGOW_HASKELL__ < 710<br />
{-# OPTIONS_GHC -fno-warn-amp #-}<br />
#endif<br />
</haskell><br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - ''sigh''<br />
<br />
With the new hierarchy, the answer would ''always'' be "use the least restrictive one".<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== '''Current stage''': Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56857Functor-Applicative-Monad Proposal2013-09-16T21:37:38Z<p>Blackout: /* Future Prelude names */ refactored 'hiding' list item</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== Future-proofing current code ==<br />
<br />
GHC 7.8 will issue two types of warnings in order to encourage wide-scale code fixing. The following describes how to get rid of them, and as a result ensures your code builds both now and after the AMP is finished.<br />
<br />
=== Missing superclasses ===<br />
<br />
(Warnings of the type "Warning: X is an instance of C, but not D")<br />
<br />
* Add Applicative/Functor instances for all your Monads. You can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
import Control.Applicative (Applicative(..))<br />
import Control.Monad (liftM, ap)<br />
<br />
-- Monad m<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadPlus. This can again be done easily using<br />
<haskell><br />
import Control.Applicative (Alternative(..))<br />
import Control.Monad (mzero, mplus)<br />
<br />
-- MonadPlus m<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
=== Future Prelude names ===<br />
<br />
"The name X clashes with a future Prelude name" - Prelude will export functions called <hask><*></hask>, <hask>join</hask> and <hask>pure</hask>, so if a module defines its own versions of them, there will be name clashes. There are multiple ways of getting rid of this type of warning (in a future-proof way).<br />
<br />
# Change your code to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
# Import Prelude definitions you need explicitly. For example, <hask>import Prelude (map, (+))</hask> would not import <hask>join</hask>, so no warning is issued as the module is compatible with the Prelude exporting <hask>join</hask>. <hask>hiding</hask>.<br />
# Due to GHC internals, you cannot use <hask>hiding (join, (<*>), pure)</hask> to silence the warnings, although this method would be future-proof. If you want to use <hask>hiding</hask>, you will have to silence the warnings using a sledgehammer <code>-fno-warn-amp</code> compiler flag. (If you do so make sure you know what you're doing, otherwise your module ''will'' break in 7.10.) To make 7.10 not complain about the then unrecognized flag, it's best to specify it in a CPP block,<br />
<haskell><br />
{-# LANGUAGE CPP #-}<br />
#if __GLASGOW_HASKELL__ >= 707 && __GLASGOW_HASKELL__ < 710<br />
{-# OPTIONS_GHC -fno-warn-amp #-}<br />
#endif<br />
</haskell><br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== '''Current stage''': Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56856Functor-Applicative-Monad Proposal2013-09-16T21:25:43Z<p>Blackout: typo</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== Future-proofing current code ==<br />
<br />
GHC 7.8 will issue two types of warnings in order to encourage wide-scale code fixing. The following describes how to get rid of them, and as a result ensures your code builds both now and after the AMP is finished.<br />
<br />
=== Missing superclasses ===<br />
<br />
(Warnings of the type "Warning: X is an instance of C, but not D")<br />
<br />
* Add Applicative/Functor instances for all your Monads. You can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
import Control.Applicative (Applicative(..))<br />
import Control.Monad (liftM, ap)<br />
<br />
-- Monad m<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadPlus. This can again be done easily using<br />
<haskell><br />
import Control.Applicative (Alternative(..))<br />
import Control.Monad (mzero, mplus)<br />
<br />
-- MonadPlus m<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
=== Future Prelude names ===<br />
<br />
"The name X clashes with a future Prelude name" - Prelude will export functions called <hask><*></hask>, <hask>join</hask> and <hask>pure</hask>, so if a module defines its own versions of them, there will be name clashes. There are multiple ways of getting rid of this type of warning.<br />
<br />
# Change your code to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
# Import Prelude definitions you need explicitly. For example, <hask>import Prelude (map, (+))</hask> would not import <hask>join</hask>, so no warning is issued as the module is compatible with the Prelude exporting <hask>join</hask>. ''Note:'' due to how GHC handles <hask>hiding</hask> statements, it impossible to make <hask>import Prelude hiding (join)</hask> not create a warning when the hidden name is not actually exported. In other words, you can't use <hask>hiding</hask>.<br />
# As a last resort, you can use the compiler option <code>-fno-warn-amp</code>, which silences all AMP warnings. Be careful with this though, because once 7.10 hits that module will break if the AMP issues haven't been fixed properly.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== '''Current stage''': Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56816Functor-Applicative-Monad Proposal2013-09-13T23:32:02Z<p>Blackout: /* Future-proofing current code */ - Brought up to date with #8004</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== Future-proofing current code ==<br />
<br />
GHC 7.8 will issue two types of warnings in order to encourage wide-scale code fixing. The following describes how to get rid of them, and as a result ensuring your code builds both now and after the AMP is finished.<br />
<br />
=== Missing superclasses ===<br />
<br />
(Warnings of the type "Warning: X is an instance of C, but not D")<br />
<br />
* Add Applicative/Functor instances for all your Monads. You can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
import Control.Applicative (Applicative(..))<br />
import Control.Monad (liftM, ap)<br />
<br />
-- Monad m<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadPlus. This can again be done easily using<br />
<haskell><br />
import Control.Applicative (Alternative(..))<br />
import Control.Monad (mzero, mplus)<br />
<br />
-- MonadPlus m<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
=== Future Prelude names ===<br />
<br />
"The name X clashes with a future Prelude name" - Prelude will export functions called <hask><*></hask>, <hask>join</hask> and <hask>pure</hask>, so if a module defines its own versions of them, there will be name clashes. There are multiple ways of getting rid of this type of warning.<br />
<br />
# Change your code to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
# Import Prelude definitions you need explicitly. For example, <hask>import Prelude (map, (+))</hask> would not import <hask>join</hask>, so no warning is issued as the module is compatible with the Prelude exporting <hask>join</hask>. ''Note:'' due to how GHC handles <hask>hiding</hask> statements, it impossible to make <hask>import Prelude hiding (join)</hask> not create a warning when the hidden name is not actually exported. In other words, you can't use <hask>hiding</hask>.<br />
# As a last resort, you can use the compiler option <code>-fno-warn-amp</code>, which silences all AMP warnings. Be careful with this though, because once 7.10 hits that module will break if the AMP issues haven't been fixed properly.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== '''Current stage''': Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=DMR&diff=56326DMR2013-06-23T13:18:53Z<p>Blackout: Redirecting to Monomorphism restriction</p>
<hr />
<div>#REDIRECT [[Monomorphism restriction]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Typeclassopedia&diff=56323Typeclassopedia2013-06-22T07:37:39Z<p>Blackout: /* Other monoidal classes: Alternative, MonadPlus, ArrowPlus */ Added explicit monoid laws for Alternative</p>
<hr />
<div>''By [[User:Byorgey|Brent Yorgey]], byorgey@cis.upenn.edu''<br />
<br />
''Originally published 12 March 2009 in [http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf issue 13] of [http://themonadreader.wordpress.com/ the Monad.Reader]. Ported to the Haskell wiki in November 2011 by [[User:Geheimdienst|Geheimdienst]].''<br />
<br />
''This is now the official version of the Typeclassopedia and supersedes the version published in the Monad.Reader. Please help update and extend it by editing it yourself or by leaving comments, suggestions, and questions on the [[Talk:Typeclassopedia|talk page]].''<br />
<br />
=Abstract=<br />
<br />
The standard Haskell libraries feature a number of type classes with algebraic or category-theoretic underpinnings. Becoming a fluent Haskell hacker requires intimate familiarity with them all, yet acquiring this familiarity often involves combing through a mountain of tutorials, blog posts, mailing list archives, and IRC logs.<br />
<br />
The goal of this document is to serve as a starting point for the student of Haskell wishing to gain a firm grasp of its standard type classes. The essentials of each type class are introduced, with examples, commentary, and extensive references for further reading.<br />
<br />
=Introduction=<br />
<br />
Have you ever had any of the following thoughts?<br />
* What the heck is a monoid, and how is it different from a mon<u>a</u>d?<br />
<br />
* I finally figured out how to use [[Parsec]] with do-notation, and someone told me I should use something called <code>Applicative</code> instead. Um, what?<br />
<br />
* Someone in the [[IRC channel|#haskell]] IRC channel used <code>(***)</code>, and when I asked lambdabot to tell me its type, it printed out scary gobbledygook that didn’t even fit on one line! Then someone used <code>fmap fmap fmap</code> and my brain exploded.<br />
<br />
* When I asked how to do something I thought was really complicated, people started typing things like <code>zip.ap fmap.(id &&& wtf)</code> and the scary thing is that they worked! Anyway, I think those people must actually be robots because there’s no way anyone could come up with that in two seconds off the top of their head.<br />
<br />
If you have, look no further! You, too, can write and understand concise, elegant, idiomatic Haskell code with the best of them.<br />
<br />
There are two keys to an expert Haskell hacker’s wisdom:<br />
# Understand the types.<br />
# Gain a deep intuition for each type class and its relationship to other type classes, backed up by familiarity with many examples.<br />
<br />
It’s impossible to overstate the importance of the first; the patient student of type signatures will uncover many profound secrets. Conversely, anyone ignorant of the types in their code is doomed to eternal uncertainty. “Hmm, it doesn’t compile ... maybe I’ll stick in an<br />
<code>fmap</code> here ... nope, let’s see ... maybe I need another <code>(.)</code> somewhere? ... um ...”<br />
<br />
The second key—gaining deep intuition, backed by examples—is also important, but much more difficult to attain. A primary goal of this document is to set you on the road to gaining such intuition. However—<br />
<br />
:''There is no royal road to Haskell. {{h:title|Well, he probably would have said it if he knew Haskell.|—Euclid}}''<br />
<br />
This document can only be a starting point, since good intuition comes from hard work, [http://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/ not from learning the right metaphor]. Anyone who reads and understands all of it will still have an arduous journey ahead—but sometimes a good starting point makes a big difference.<br />
<br />
It should be noted that this is not a Haskell tutorial; it is assumed that the reader is already familiar with the basics of Haskell, including the standard <code>[http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html Prelude]</code>, the type system, data types, and type classes.<br />
<br />
The type classes we will be discussing and their interrelationships:<br />
<br />
[[Image:Typeclassopedia-diagram.png]]<br />
<br />
{{note|<code>Semigroup</code> can be found in the [http://hackage.haskell.org/package/semigroups <code>semigroups</code> package], <code>Apply</code> in the [http://hackage.haskell.org/package/semigroupoids <code>semigroupoids</code> package], and <code>Comonad</code> in the [http://hackage.haskell.org/package/comonad <code>comonad</code> package].}}<br />
<br />
* <span style="border-bottom: 2px solid black">Solid arrows</span> point from the general to the specific; that is, if there is an arrow from <code>Foo</code> to <code>Bar</code> it means that every <code>Bar</code> is (or should be, or can be made into) a <code>Foo</code>.<br />
* <span style="border-bottom: 2px dotted black">Dotted arrows</span> indicate some other sort of relationship.<br />
* <code>Monad</code> and <code>ArrowApply</code> are equivalent.<br />
* <code>Semigroup</code>, <code>Apply</code> and <code>Comonad</code> are greyed out since they are not actually (yet?) in the standard Haskell libraries {{noteref}}.<br />
<br />
One more note before we begin. The original spelling of “type class” is with two words, as evidenced by, for example, the [http://haskell.org/onlinereport/ Haskell 98 Revised Report], early papers on type classes like [http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.103.5639 Type classes in Haskell] and [http://research.microsoft.com/en-us/um/people/simonpj/papers/type-class-design-space/ Type classes: exploring the design space], and [http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.168.4008 Hudak et al.’s history of Haskell]. However, as often happens with two-word phrases that see a lot of use, it has started to show up as one word (“typeclass”) or, rarely, hyphenated (“type-class”). When wearing my prescriptivist hat, I prefer “type class”, but realize (after changing into my descriptivist hat) that there's probably not much I can do about it.<br />
<br />
We now begin with the simplest type class of all: <code>Functor</code>.<br />
<br />
=Functor=<br />
<br />
The <code>Functor</code> class ([http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#t:Functor haddock]) is the most basic and ubiquitous type class in the Haskell libraries. A simple intuition is that a <code>Functor</code> represents a “container” of some sort, along with the ability to apply a function uniformly to every element in the container. For example, a list is a container of elements, and we can apply a function to every element of a list, using <code>map</code>. As another example, a binary tree is also a container of elements, and it’s not hard to come up with a way to recursively apply a function to every element in a tree.<br />
<br />
Another intuition is that a <code>Functor</code> represents some sort of “computational context”. This intuition is generally more useful, but is more difficult to explain, precisely because it is so general. Some examples later should help to clarify the <code>Functor</code>-as-context point of view.<br />
<br />
In the end, however, a <code>Functor</code> is simply what it is defined to be; doubtless there are many examples of <code>Functor</code> instances that don’t exactly fit either of the above intuitions. The wise student will focus their attention on definitions and examples, without leaning too heavily on any particular metaphor. Intuition will come, in time, on its own.<br />
<br />
==Definition==<br />
<br />
Here is the type class declaration for <code>Functor</code>:<br />
<br />
<haskell><br />
class Functor f where<br />
fmap :: (a -> b) -> f a -> f b<br />
</haskell><br />
<br />
<code>Functor</code> is exported by the <code>Prelude</code>, so no special imports are needed to use it.<br />
<br />
First, the <code>f a</code> and <code>f b</code> in the type signature for <code>fmap</code> tell us that <code>f</code> isn’t just a type; it is a ''type constructor'' which takes another type as a parameter. (A more precise way to say this is that the ''kind'' of <code>f</code> must be <code>* -> *</code>.) For example, <code>Maybe</code> is such a type constructor: <code>Maybe</code> is not a type in and of itself, but requires another type as a parameter, like <code>Maybe Integer</code>. So it would not make sense to say <code>instance Functor Integer</code>, but it could make sense to say <code>instance Functor Maybe</code>.<br />
<br />
Now look at the type of <code>fmap</code>: it takes any function from <code>a</code> to <code>b</code>, and a value of type <code>f a</code>, and outputs a value of type <code>f b</code>. From the container point of view, the intention is that <code>fmap</code> applies a function to each element of a container, without altering the structure of the container. From the context point of view, the intention is that <code>fmap</code> applies a function to a value without altering its context. Let’s look at a few specific examples.<br />
<br />
==Instances==<br />
<br />
{{note|Recall that <code>[]</code> has two meanings in Haskell: it can either stand for the empty list, or, as here, it can represent the list type constructor (pronounced “list-of”). In other words, the type <code>[a]</code> (list-of-<code>a</code>) can also be written <code>[] a</code>.}}<br />
<br />
{{note|You might ask why we need a separate <code>map</code> function. Why not just do away with the current list-only <code>map</code> function, and rename <code>fmap</code> to <code>map</code> instead? Well, that’s a good question. The usual argument is that someone just learning Haskell, when using <code>map</code> incorrectly, would much rather see an error about lists than about <code>Functor</code>s.}}<br />
<br />
As noted before, the list constructor <code>[]</code> is a functor {{noteref}}; we can use the standard list function <code>map</code> to apply a function to each element of a list {{noteref}}. The <code>Maybe</code> type constructor is also a functor, representing a container which might hold a single element. The function <code>fmap g</code> has no effect on <code>Nothing</code> (there are no elements to which <code>g</code> can be applied), and simply applies <code>g</code> to the single element inside a <code>Just</code>. Alternatively, under the context interpretation, the list functor represents a context of nondeterministic choice; that is, a list can be thought of as representing a single value which is nondeterministically chosen from among several possibilities (the elements of the list). Likewise, the <code>Maybe</code> functor represents a context with possible failure. These instances are:<br />
<br />
<haskell><br />
instance Functor [] where<br />
fmap _ [] = []<br />
fmap g (x:xs) = g x : fmap g xs<br />
-- or we could just say fmap = map<br />
<br />
instance Functor Maybe where<br />
fmap _ Nothing = Nothing<br />
fmap g (Just a) = Just (g a)<br />
</haskell><br />
<br />
As an aside, in idiomatic Haskell code you will often see the letter <code>f</code> used to stand for both an arbitrary <code>Functor</code> and an arbitrary function. In this document, <code>f</code> represents only <code>Functor</code>s, and <code>g</code> or <code>h</code> always represent functions, but you should be aware of the potential confusion. In practice, what <code>f</code> stands for should always be clear from the context, by noting whether it is part of a type or part of the code.<br />
<br />
There are other <code>Functor</code> instances in the standard libraries; below are a few. Note that some of these instances are not exported by the <code>Prelude</code>; to access them, you can import <code>Control.Monad.Instances</code>.<br />
<br />
* <code>Either e</code> is an instance of <code>Functor</code>; <code>Either e a</code> represents a container which can contain either a value of type <code>a</code>, or a value of type <code>e</code> (often representing some sort of error condition). It is similar to <code>Maybe</code> in that it represents possible failure, but it can carry some extra information about the failure as well.<br />
<br />
* <code>((,) e)</code> represents a container which holds an “annotation” of type <code>e</code> along with the actual value it holds. It might be clearer to write it as <code>(e,)</code>, by analogy with an operator section like <code>(1+)</code>, but that syntax is not allowed in types (although it is allowed in expressions with the <code>TupleSections</code> extension enabled). However, you can certainly ''think'' of it as <code>(e,)</code>.<br />
<br />
* <code>((->) e)</code> (which can be thought of as <code>(e ->)</code>; see above), the type of functions which take a value of type <code>e</code> as a parameter, is a <code>Functor</code>. As a container, <code>(e -> a)</code> represents a (possibly infinite) set of values of <code>a</code>, indexed by values of <code>e</code>. Alternatively, and more usefully, <code>((->) e)</code> can be thought of as a context in which a value of type <code>e</code> is available to be consulted in a read-only fashion. This is also why <code>((->) e)</code> is sometimes referred to as the ''reader monad''; more on this later.<br />
<br />
* <code>IO</code> is a <code>Functor</code>; a value of type <code>IO a</code> represents a computation producing a value of type <code>a</code> which may have I/O effects. If <code>m</code> computes the value <code>x</code> while producing some I/O effects, then <code>fmap g m</code> will compute the value <code>g x</code> while producing the same I/O effects.<br />
<br />
* Many standard types from the [http://hackage.haskell.org/package/containers/ containers library] (such as <code>Tree</code>, <code>Map</code>, and <code>Sequence</code>) are instances of <code>Functor</code>. A notable exception is <code>Set</code>, which cannot be made a <code>Functor</code> in Haskell (although it is certainly a mathematical functor) since it requires an <code>Ord</code> constraint on its elements; <code>fmap</code> must be applicable to ''any'' types <code>a</code> and <code>b</code>. However, <code>Set</code> (and other similarly restricted data types) can be made an instance of a suitable generalization of <code>Functor</code>, either by [http://article.gmane.org/gmane.comp.lang.haskell.cafe/78052/ making <code>a</code> and <code>b</code> arguments to the <code>Functor</code> type class themselves], or by adding an [http://blog.omega-prime.co.uk/?p=127 associated constraint].<br />
<br />
{{Exercises|<br />
<ol><br />
<li>Implement <code>Functor</code> instances for <code>Either e</code> and <code>((->) e)</code>.</li><br />
<li>Implement <code>Functor</code> instances for <code>((,) e)</code> and for <code>Pair</code>, defined as <br />
<br />
<haskell>data Pair a = Pair a a</haskell><br />
<br />
Explain their similarities and differences.<br />
</li><br />
<li>Implement a <code>Functor</code> instance for the type <code>ITree</code>, defined as<br />
<br />
<haskell><br />
data ITree a = Leaf (Int -> a) <br />
| Node [ITree a]<br />
</haskell><br />
</li><br />
<li>Give an example of a type of kind <code>* -> *</code> which cannot be made an instance of <code>Functor</code> (without using <code>undefined</code>).<br />
</li><br />
<li>Is this statement true or false? <br />
<br />
:''The composition of two <code>Functor</code>s is also a <code>Functor</code>.''<br />
<br />
If false, give a counterexample; if true, prove it by exhibiting some appropriate Haskell code.<br />
</li><br />
</ol><br />
}}<br />
<br />
==Laws==<br />
<br />
As far as the Haskell language itself is concerned, the only requirement to be a <code>Functor</code> is an implementation of <code>fmap</code> with the proper type. Any sensible <code>Functor</code> instance, however, will also satisfy the ''functor laws'', which are part of the definition of a mathematical functor. There are two:<br />
<br />
<haskell><br />
fmap id = id<br />
fmap (g . h) = (fmap g) . (fmap h)<br />
</haskell><br />
<br />
{{note|Technically, these laws make <code>f</code> and <code>fmap</code> together an endofunctor on ''Hask'', the category of Haskell types (ignoring [[Bottom|&perp;]], which is a party pooper). See [http://en.wikibooks.org/wiki/Haskell/Category_theory Wikibook: Category theory].}}<br />
<br />
Together, these laws ensure that <code>fmap g</code> does not change the ''structure'' of a container, only the elements. Equivalently, and more simply, they ensure that <code>fmap g</code> changes a value without altering its context {{noteref}}.<br />
<br />
The first law says that mapping the identity function over every item in a container has no effect. The second says that mapping a composition of two functions over every item in a container is the same as first mapping one function, and then mapping the other.<br />
<br />
As an example, the following code is a “valid” instance of <code>Functor</code> (it typechecks), but it violates the functor laws. Do you see why?<br />
<br />
<haskell><br />
-- Evil Functor instance<br />
instance Functor [] where<br />
fmap _ [] = []<br />
fmap g (x:xs) = g x : g x : fmap g xs<br />
</haskell><br />
<br />
Any Haskeller worth their salt would reject this code as a gruesome abomination.<br />
<br />
Unlike some other type classes we will encounter, a given type has at most one valid instance of <code>Functor</code>. This [http://article.gmane.org/gmane.comp.lang.haskell.libraries/15384 can be proven] via the [http://homepages.inf.ed.ac.uk/wadler/topics/parametricity.html#free ''free theorem''] for the type of <code>fmap</code>. In fact, [http://byorgey.wordpress.com/2010/03/03/deriving-pleasure-from-ghc-6-12-1/ GHC can automatically derive] <code>Functor</code> instances for many data types.<br />
<br />
A similar argument also shows that any <code>Functor</code> instance satisfying the first law (<code>fmap id = id</code>) will automatically satisfy the second law as well. Practically, this means that only the first law needs to be checked (usually by a very straightforward induction) to ensure that a <code>Functor</code> instance is valid.<br />
<br />
{{Exercises|<br />
# Although it is not possible for a <code>Functor</code> instance to satisfy the first <code>Functor</code> law but not the second, the reverse is possible. Give an example of a (bogus) <code>Functor</code> instance which satisfies the second law but not the first.<br />
# Which laws are violated by the evil <code>Functor</code> instance for list shown above: both laws, or the first law alone? Give specific counterexamples.<br />
}}<br />
<br />
==Intuition==<br />
<br />
There are two fundamental ways to think about <code>fmap</code>. The first has already been mentioned: it takes two parameters, a function and a container, and applies the function “inside” the container, producing a new container. Alternately, we can think of <code>fmap</code> as applying a function to a value in a context (without altering the context).<br />
<br />
Just like all other Haskell functions of “more than one parameter”, however, <code>fmap</code> is actually ''curried'': it does not really take two parameters, but takes a single parameter and returns a function. For emphasis, we can write <code>fmap</code>’s type with extra parentheses: <code>fmap :: (a -> b) -> (f a -> f b)</code>. Written in this form, it is apparent that <code>fmap</code> transforms a “normal” function (<code>g :: a -> b</code>) into one which operates over containers/contexts (<code>fmap g :: f a -> f b</code>). This transformation is often referred to as a ''lift''; <code>fmap</code> “lifts” a function from the “normal world” into the “<code>f</code> world”.<br />
<br />
==Further reading==<br />
<br />
A good starting point for reading about the category theory behind the concept of a functor is the excellent [http://en.wikibooks.org/wiki/Haskell/Category_theory Haskell wikibook page on category theory].<br />
<br />
=Applicative=<br />
<br />
A somewhat newer addition to the pantheon of standard Haskell type classes, ''applicative functors'' represent an abstraction lying in between <code>Functor</code> and <code>Monad</code> in expressivity, first described by McBride and Paterson. The title of their classic paper, [http://www.soi.city.ac.uk/~ross/papers/Applicative.html Applicative Programming with Effects], gives a hint at the intended intuition behind the [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html <code>Applicative</code>] type class. It encapsulates certain sorts of “effectful” computations in a functionally pure way, and encourages an “applicative” programming style. Exactly what these things mean will be seen later.<br />
<br />
==Definition==<br />
<br />
Recall that <code>Functor</code> allows us to lift a “normal” function to a function on computational contexts. But <code>fmap</code> doesn’t allow us to apply a function which is itself in a context to a value in a context. <code>Applicative</code> gives us just such a tool, <code>(<*>)</code>. It also provides a method, <code>pure</code>, for embedding values in a default, “effect free” context. Here is the type class declaration for <code>Applicative</code>, as defined in <code>Control.Applicative</code>:<br />
<br />
<haskell><br />
class Functor f => Applicative f where<br />
pure :: a -> f a<br />
(<*>) :: f (a -> b) -> f a -> f b<br />
</haskell><br />
<br />
Note that every <code>Applicative</code> must also be a <code>Functor</code>. In fact, as we will see, <code>fmap</code> can be implemented using the <code>Applicative</code> methods, so every <code>Applicative</code> is a functor whether we like it or not; the <code>Functor</code> constraint forces us to be honest.<br />
<br />
{{note|Recall that <code>($)</code> is just function application: <code>f $ x {{=}} f x</code>.}}<br />
<br />
As always, it’s crucial to understand the type signatures. First, consider <code>(<*>)</code>: the best way of thinking about it comes from noting that the type of <code>(<*>)</code> is similar to the type of <code>($)</code> {{noteref}}, but with everything enclosed in an <code>f</code>. In other words, <code>(<*>)</code> is just function application within a computational context. The type of <code>(<*>)</code> is also very similar to the type of <code>fmap</code>; the only difference is that the first parameter is <code>f (a -> b)</code>, a function in a context, instead of a “normal” function <code>(a -> b)</code>.<br />
<br />
<code>pure</code> takes a value of any type <code>a</code>, and returns a context/container of type <code>f a</code>. The intention is that <code>pure</code> creates some sort of “default” container or “effect free” context. In fact, the behavior of <code>pure</code> is quite constrained by the laws it should satisfy in conjunction with <code>(<*>)</code>. Usually, for a given implementation of <code>(<*>)</code> there is only one possible implementation of <code>pure</code>.<br />
<br />
(Note that previous versions of the Typeclassopedia explained <code>pure</code> in terms of a type class <code>Pointed</code>, which can still be found in the [http://hackage.haskell.org/package/pointed <code>pointed</code> package]. However, the current consensus is that <code>Pointed</code> is not very useful after all. For a more detailed explanation, see [[Why not Pointed?]])<br />
<br />
==Laws==<br />
<br />
{{note|See<br />
[http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html haddock for Applicative] and [http://www.soi.city.ac.uk/~ross/papers/Applicative.html Applicative programming with effects]}}<br />
<br />
Traditionally, there are four laws that <code>Applicative</code> instances should satisfy {{noteref}}. In some sense, they are all concerned with making sure that <code>pure</code> deserves its name:<br />
<br />
* The identity law:<br /><haskell>pure id <*> v = v</haskell><br />
* Homomorphism:<br /><haskell>pure f <*> pure x = pure (f x)</haskell>Intuitively, applying a non-effectful function to a non-effectful argument in an effectful context is the same as just applying the function to the argument and then injecting the result into the context with <code>pure</code>.<br />
* Interchange:<br /><haskell>u <*> pure y = pure ($ y) <*> u</haskell>Intuitively, this says that when evaluating the application of an effectful function to a pure argument, the order in which we evaluate the function and its argument doesn't matter.<br />
* Composition:<br /><haskell>u <*> (v <*> w) = pure (.) <*> u <*> v <*> w </haskell>This one is the trickiest law to gain intuition for. In some sense it is expressing a sort of associativity property of <code>(<*>)</code>. The reader may wish to simply convince themselves that this law is type-correct.<br />
<br />
Considered as left-to-right rewrite rules, the homomorphism, interchange, and composition laws actually constitute an algorithm for transforming any expression using <code>pure</code> and <code>(<*>)</code> into a canonical form with only a single use of <code>pure</code> at the very beginning and only left-nested occurrences of <code>(<*>)</code>. Composition allows reassociating <code>(<*>)</code>; interchange allows moving occurrences of <code>pure</code> leftwards; and homomorphism allows collapsing multiple adjacent occurrences of <code>pure</code> into one.<br />
<br />
There is also a law specifying how <code>Applicative</code> should relate to <code>Functor</code>:<br />
<br />
<haskell><br />
fmap g x = pure g <*> x<br />
</haskell><br />
<br />
It says that mapping a pure function <code>g</code> over a context <code>x</code> is the same as first injecting <code>g</code> into a context with <code>pure</code>, and then applying it to <code>x</code> with <code>(<*>)</code>. In other words, we can decompose <code>fmap</code> into two more atomic operations: injection into a context, and application within a context. The <code>Control.Applicative</code> module also defines <code>(<$>)</code> as a synonym for <code>fmap</code>, so the above law can also be expressed as:<br />
<br />
<code>g <$> x = pure g <*> x</code>.<br />
<br />
{{Exercises|<br />
# (Tricky) One might imagine a variant of the interchange law that says something about applying a pure function to an effectful argument. Using the above laws, prove that<haskell>pure f <*> x = pure (flip ($)) <*> x <*> pure f</haskell><br />
}}<br />
<br />
==Instances==<br />
<br />
Most of the standard types which are instances of <code>Functor</code> are also instances of <code>Applicative</code>.<br />
<br />
<code>Maybe</code> can easily be made an instance of <code>Applicative</code>; writing such an instance is left as an exercise for the reader.<br />
<br />
The list type constructor <code>[]</code> can actually be made an instance of <code>Applicative</code> in two ways; essentially, it comes down to whether we want to think of lists as ordered collections of elements, or as contexts representing multiple results of a nondeterministic computation (see Wadler’s [http://www.springerlink.com/content/y7450255v2670167/ How to replace failure by a list of successes]).<br />
<br />
Let’s first consider the collection point of view. Since there can only be one instance of a given type class for any particular type, one or both of the list instances of <code>Applicative</code> need to be defined for a <code>newtype</code> wrapper; as it happens, the nondeterministic computation instance is the default, and the collection instance is defined in terms of a <code>newtype</code> called <code>ZipList</code>. This instance is:<br />
<br />
<haskell><br />
newtype ZipList a = ZipList { getZipList :: [a] }<br />
<br />
instance Applicative ZipList where<br />
pure = undefined -- exercise<br />
(ZipList gs) <*> (ZipList xs) = ZipList (zipWith ($) gs xs)<br />
</haskell><br />
<br />
To apply a list of functions to a list of inputs with <code>(<*>)</code>, we just match up the functions and inputs elementwise, and produce a list of the resulting outputs. In other words, we “zip” the lists together with function application, <code>($)</code>; hence the name <code>ZipList</code>. <br />
<br />
The other <code>Applicative</code> instance for lists, based on the nondeterministic computation point of view, is:<br />
<br />
<haskell><br />
instance Applicative [] where<br />
pure x = [x]<br />
gs <*> xs = [ g x | g <- gs, x <- xs ]<br />
</haskell><br />
<br />
Instead of applying functions to inputs pairwise, we apply each function to all the inputs in turn, and collect all the results in a list.<br />
<br />
Now we can write nondeterministic computations in a natural style. To add the numbers <code>3</code> and <code>4</code> deterministically, we can of course write <code>(+) 3 4</code>. But suppose instead of <code>3</code> we have a nondeterministic computation that might result in <code>2</code>, <code>3</code>, or <code>4</code>; then we can write<br />
<br />
<haskell><br />
pure (+) <*> [2,3,4] <*> pure 4<br />
</haskell><br />
<br />
or, more idiomatically,<br />
<br />
<haskell><br />
(+) <$> [2,3,4] <*> pure 4.<br />
</haskell><br />
<br />
There are several other <code>Applicative</code> instances as well:<br />
<br />
* <code>IO</code> is an instance of <code>Applicative</code>, and behaves exactly as you would think: to execute <code>m1 <*> m2</code>, first <code>m1</code> is executed, resulting in a function <code>f</code>, then <code>m2</code> is executed, resulting in a value <code>x</code>, and finally the value <code>f x</code> is returned as the result of executing <code>m1 <*> m2</code>.<br />
<br />
* <code>((,) a)</code> is an <code>Applicative</code>, as long as <code>a</code> is an instance of <code>Monoid</code> ([[#Monoid|section Monoid]]). The <code>a</code> values are accumulated in parallel with the computation.<br />
<br />
* The <code>Applicative</code> module defines the <code>Const</code> type constructor; a value of type <code>Const a b</code> simply contains an <code>a</code>. This is an instance of <code>Applicative</code> for any <code>Monoid a</code>; this instance becomes especially useful in conjunction with things like <code>Foldable</code> ([[#Foldable|section Foldable]]).<br />
<br />
* The <code>WrappedMonad</code> and <code>WrappedArrow</code> newtypes make any instances of <code>Monad</code> ([[#Monad|section Monad]]) or <code>Arrow</code> ([[#Arrow|section Arrow]]) respectively into instances of <code>Applicative</code>; as we will see when we study those type classes, both are strictly more expressive than <code>Applicative</code>, in the sense that the <code>Applicative</code> methods can be implemented in terms of their methods.<br />
<br />
{{Exercises|<br />
# Implement an instance of <code>Applicative</code> for <code>Maybe</code>.<br />
# Determine the correct definition of <code>pure</code> for the <code>ZipList</code> instance of <code>Applicative</code>—there is only one implementation that satisfies the law relating <code>pure</code> and <code>(<*>)</code>.<br />
}}<br />
<br />
==Intuition==<br />
<br />
McBride and Paterson’s paper introduces the notation <math>[[g \; x_1 \; x_2 \; \cdots \; x_n]]\ </math> to denote function application in a computational context. If each <math>x_i\ </math> has type <math>f \; t_i\ </math> for some applicative functor <math>f\ </math>, and <math>g\ </math> has type <math>t_1 \to t_2 \to \dots \to t_n \to t\ </math>, then the entire expression <math>[[g \; x_1 \; \cdots \; x_n]]\ </math> has type <math>f \; t\ </math>. You can think of this as applying a function to multiple “effectful” arguments. In this sense, the double bracket notation is a generalization of <code>fmap</code>, which allows us to apply a function to a single argument in a context.<br />
<br />
Why do we need <code>Applicative</code> to implement this generalization of <code>fmap</code>? Suppose we use <code>fmap</code> to apply <code>g</code> to the first parameter <code>x1</code>. Then we get something of type <code>f (t2 -> ... t)</code>, but now we are stuck: we can’t apply this function-in-a-context to the next argument with <code>fmap</code>. However, this is precisely what <code>(<*>)</code> allows us to do.<br />
<br />
This suggests the proper translation of the idealized notation <math>[[g \; x_1 \; x_2 \; \cdots \; x_n]]\ </math> into Haskell, namely<br />
<haskell><br />
g <$> x1 <*> x2 <*> ... <*> xn,<br />
</haskell><br />
<br />
recalling that <code>Control.Applicative</code> defines <code>(<$>)</code> as convenient infix shorthand for <code>fmap</code>. This is what is meant by an “applicative style”—effectful computations can still be described in terms of function application; the only difference is that we have to use the special operator <code>(<*>)</code> for application instead of simple juxtaposition.<br />
<br />
Note that <code>pure</code> allows embedding “non-effectful” arguments in the middle of an idiomatic application, like<br />
<haskell><br />
g <$> x1 <*> pure x2 <*> x3<br />
</haskell><br />
which has type <code>f d</code>, given<br />
<haskell><br />
g :: a -> b -> c -> d<br />
x1 :: f a<br />
x2 :: b<br />
x3 :: f c<br />
</haskell><br />
<br />
The double brackets are commonly known as “idiom brackets”, because they allow writing “idiomatic” function application, that is, function application that looks normal but has some special, non-standard meaning (determined by the particular instance of <code>Applicative</code> being used). Idiom brackets are not supported by GHC, but they are supported by the [http://personal.cis.strath.ac.uk/~conor/pub/she/ Strathclyde Haskell Enhancement], a preprocessor which (among many other things) translates idiom brackets into standard uses of <code>(<$>)</code> and <code>(<*>)</code>. This can result in much more readable code when making heavy use of <code>Applicative</code>.<br />
<br />
==Alternative formulation==<br />
<br />
An alternative, equivalent formulation of <code>Applicative</code> is given by<br />
<br />
<haskell><br />
class Functor f => Monoidal f where<br />
unit :: f ()<br />
(**) :: f a -> f b -> f (a,b)<br />
</haskell><br />
<br />
Intuitively, this states that a <i>monoidal</i> functor is one which has some sort of "default shape" and which supports some sort of "combining" operation. <code>pure</code> and <code>(<*>)</code> are equivalent in power to <code>unit</code> and <code>(**)</code> (see the Exercises below).<br />
<br />
Furthermore, to deserve the name "monoidal" (see the [[#Monoid|section on Monoids]]), instances of <code>Monoidal</code> ought to satisfy the following laws, which seem much more straightforward than the traditional <code>Applicative</code> laws:<br />
<br />
{{note|Here <code>g *** h {{=}} \(x,y) -> (g x, h y)</code>. See [[#Arrow|Arrows]].}}<br />
* Naturality{{noteref}}: <haskell>fmap (g *** h) (u ** v) = fmap g u ** fmap h v</haskell><br />
{{note|In this and the following laws, <code>≅</code> refers to <i>isomorphism</i> rather than equality. In particular we consider <code>(x,()) ≅ x ≅ ((),x)</code> and <code>((x,y),z) ≅ (x,(y,z))</code>.}}<br />
* Left identity{{noteref}}: <haskell>unit ** v ≅ v</haskell><br />
* Right identity: <haskell>u ** unit ≅ u</haskell><br />
* Associativity: <haskell>u ** (v ** w) ≅ (u ** v) ** w</haskell><br />
<br />
These turn out to be equivalent to the usual <code>Applicative</code> laws.<br />
<br />
Much of this section was taken from [http://blog.ezyang.com/2012/08/applicative-functors/ a blog post by Edward Z. Yang]; see his actual post for a bit more information.<br />
<br />
{{Exercises|<br />
# Implement <code>pure</code> and <code>(<*>)</code> in terms of <code>unit</code> and <code>(**)</code>, and vice versa.<br />
# (Tricky) Prove that given your implementations from the previous exercise, the usual <code>Applicative</code> laws and the <code>Monoidal</code> laws stated above are equivalent.<br />
}}<br />
<br />
==Further reading==<br />
<br />
There are many other useful combinators in the standard libraries implemented in terms of <code>pure</code> and <code>(<*>)</code>: for example, <code>(*>)</code>, <code>(<*)</code>, <code>(<**>)</code>, <code>(<$)</code>, and so on (see [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html haddock for Applicative]). Judicious use of such secondary combinators can often make code using <code>Applicative</code>s much easier to read.<br />
<br />
[http://www.soi.city.ac.uk/~ross/papers/Applicative.html McBride and Paterson’s original paper] is a treasure-trove of information and examples, as well as some perspectives on the connection between <code>Applicative</code> and category theory. Beginners will find it difficult to make it through the entire paper, but it is extremely well-motivated—even beginners will be able to glean something from reading as far as they are able.<br />
<br />
{{note|Introduced by [http://conal.net/papers/simply-reactive/ an earlier paper] that was since superseded by [http://conal.net/papers/push-pull-frp/ Push-pull functional reactive programming].}}<br />
<br />
Conal Elliott has been one of the biggest proponents of <code>Applicative</code>. For example, the [http://conal.net/papers/functional-images/ Pan library for functional images] and the reactive library for functional reactive programming (FRP) {{noteref}} make key use of it; his blog also contains [http://conal.net/blog/tag/applicative-functor many examples of <code>Applicative</code> in action]. Building on the work of McBride and Paterson, Elliott also built the [[TypeCompose]] library, which embodies the observation (among others) that <code>Applicative</code> types are closed under composition; therefore, <code>Applicative</code> instances can often be automatically derived for complex types built out of simpler ones.<br />
<br />
Although the [http://hackage.haskell.org/package/parsec Parsec parsing library] ([http://legacy.cs.uu.nl/daan/download/papers/parsec-paper.pdf paper]) was originally designed for use as a monad, in its most common use cases an <code>Applicative</code> instance can be used to great effect; [http://www.serpentine.com/blog/2008/02/06/the-basics-of-applicative-functors-put-to-practical-work/ Bryan O’Sullivan’s blog post] is a good starting point. If the extra power provided by <code>Monad</code> isn’t needed, it’s usually a good idea to use <code>Applicative</code> instead.<br />
<br />
A couple other nice examples of <code>Applicative</code> in action include the [http://chrisdone.com/blog/html/2009-02-10-applicative-configfile-hsql.html ConfigFile and HSQL libraries] and the [http://groups.inf.ed.ac.uk/links/formlets/ formlets library].<br />
<br />
Gershom Bazerman's [http://comonad.com/reader/2012/abstracting-with-applicatives/ post] contains many insights into applicatives.<br />
<br />
=Monad=<br />
<br />
It’s a safe bet that if you’re reading this, you’ve heard of monads—although it’s quite possible you’ve never heard of <code>Applicative</code> before, or <code>Arrow</code>, or even <code>Monoid</code>. Why are monads such a big deal in Haskell? There are several reasons.<br />
<br />
* Haskell does, in fact, single out monads for special attention by making them the framework in which to construct I/O operations.<br />
* Haskell also singles out monads for special attention by providing a special syntactic sugar for monadic expressions: the <code>do</code>-notation.<br />
* <code>Monad</code> has been around longer than other abstract models of computation such as <code>Applicative</code> or <code>Arrow</code>.<br />
* The more monad tutorials there are, the harder people think monads must be, and the more new monad tutorials are written by people who think they finally “get” monads (the [http://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/ monad tutorial fallacy]).<br />
<br />
I will let you judge for yourself whether these are good reasons.<br />
<br />
In the end, despite all the hoopla, <code>Monad</code> is just another type class. Let’s take a look at its definition.<br />
<br />
==Definition==<br />
<br />
The type class declaration for [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#t:Monad <code>Monad</code>] is:<br />
<br />
<haskell><br />
class Monad m where<br />
return :: a -> m a<br />
(>>=) :: m a -> (a -> m b) -> m b<br />
(>>) :: m a -> m b -> m b<br />
m >> n = m >>= \_ -> n<br />
<br />
fail :: String -> m a<br />
</haskell><br />
<br />
The <code>Monad</code> type class is exported by the <code>Prelude</code>, along with a few standard instances. However, many utility functions are found in [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad.html <code>Control.Monad</code>], and there are also several instances (such as <code>((->) e)</code>) defined in [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-Instances.html <code>Control.Monad.Instances</code>].<br />
<br />
Let’s examine the methods in the <code>Monad</code> class one by one. The type of <code>return</code> should look familiar; it’s the same as <code>pure</code>. Indeed, <code>return</code> ''is'' <code>pure</code>, but with an unfortunate name. (Unfortunate, since someone coming from an imperative programming background might think that <code>return</code> is like the C or Java keyword of the same name, when in fact the similarities are minimal.) From a mathematical point of view, every monad is an applicative functor, but for historical reasons, the <code>Monad</code> type class declaration unfortunately does not require this.<br />
<br />
We can see that <code>(>>)</code> is a specialized version of <code>(>>=)</code>, with a default implementation given. It is only included in the type class declaration so that specific instances of <code>Monad</code> can override the default implementation of <code>(>>)</code> with a more efficient one, if desired. Also, note that although <code>_ >> n = n</code> would be a type-correct implementation of <code>(>>)</code>, it would not correspond to the intended semantics: the intention is that <code>m >> n</code> ignores the ''result'' of <code>m</code>, but not its ''effects''.<br />
<br />
The <code>fail</code> function is an awful hack that has no place in the <code>Monad</code> class; more on this later.<br />
<br />
The only really interesting thing to look at—and what makes <code>Monad</code> strictly more powerful than <code>Applicative</code>—is <code>(>>=)</code>, which is often called ''bind''. An alternative definition of <code>Monad</code> could look like:<br />
<br />
<haskell><br />
class Applicative m => Monad' m where<br />
(>>=) :: m a -> (a -> m b) -> m b<br />
</haskell><br />
<br />
We could spend a while talking about the intuition behind <code>(>>=)</code>—and we will. But first, let’s look at some examples.<br />
<br />
==Instances==<br />
<br />
Even if you don’t understand the intuition behind the <code>Monad</code> class, you can still create instances of it by just seeing where the types lead you. You may be surprised to find that this actually gets you a long way towards understanding the intuition; at the very least, it will give you some concrete examples to play with as you read more about the <code>Monad</code> class in general. The first few examples are from the standard <code>Prelude</code>; the remaining examples are from the [http://hackage.haskell.org/package/transformers <code>transformers</code> package].<br />
<br />
<ul><br />
<li>The simplest possible instance of <code>Monad</code> is [http://hackage.haskell.org/packages/archive/mtl/1.1.0.2/doc/html/Control-Monad-Identity.html <code>Identity</code>], which is described in Dan Piponi’s highly recommended blog post on [http://blog.sigfpe.com/2007/04/trivial-monad.html The Trivial Monad]. Despite being “trivial”, it is a great introduction to the <code>Monad</code> type class, and contains some good exercises to get your brain working.<br />
</li><br />
<li>The next simplest instance of <code>Monad</code> is <code>Maybe</code>. We already know how to write <code>return</code>/<code>pure</code> for <code>Maybe</code>. So how do we write <code>(>>=)</code>? Well, let’s think about its type. Specializing for <code>Maybe</code>, we have<br />
<br />
<haskell><br />
(>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b.<br />
</haskell><br />
<br />
If the first argument to <code>(>>=)</code> is <code>Just x</code>, then we have something of type <code>a</code> (namely, <code>x</code>), to which we can apply the second argument—resulting in a <code>Maybe b</code>, which is exactly what we wanted. What if the first argument to <code>(>>=)</code> is <code>Nothing</code>? In that case, we don’t have anything to which we can apply the <code>a -> Maybe b</code> function, so there’s only one thing we can do: yield <code>Nothing</code>. This instance is:<br />
<br />
<haskell><br />
instance Monad Maybe where<br />
return = Just<br />
(Just x) >>= g = g x<br />
Nothing >>= _ = Nothing<br />
</haskell><br />
<br />
We can already get a bit of intuition as to what is going on here: if we build up a computation by chaining together a bunch of functions with <code>(>>=)</code>, as soon as any one of them fails, the entire computation will fail (because <code>Nothing >>= f</code> is <code>Nothing</code>, no matter what <code>f</code> is). The entire computation succeeds only if all the constituent functions individually succeed. So the <code>Maybe</code> monad models computations which may fail.<br />
</li><br />
<br />
<li>The <code>Monad</code> instance for the list constructor <code>[]</code> is similar to its <code>Applicative</code> instance; see the exercise below.<br />
</li><br />
<br />
<li>Of course, the <code>IO</code> constructor is famously a <code>Monad</code>, but its implementation is somewhat magical, and may in fact differ from compiler to compiler. It is worth emphasizing that the <code>IO</code> monad is the ''only'' monad which is magical. It allows us to build up, in an entirely pure way, values representing possibly effectful computations. The special value <code>main</code>, of type <code>IO ()</code>, is taken by the runtime and actually executed, producing actual effects. Every other monad is functionally pure, and requires no special compiler support. We often speak of monadic values as “effectful computations”, but this is because some monads allow us to write code ''as if'' it has side effects, when in fact the monad is hiding the plumbing which allows these apparent side effects to be implemented in a functionally pure way.<br />
</li><br />
<br />
<li>As mentioned earlier, <code>((->) e)</code> is known as the ''reader monad'', since it describes computations in which a value of type <code>e</code> is available as a read-only environment.<br />
<br />
The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Reader.html <code>Control.Monad.Reader</code>] module provides the <code>Reader e a</code> type, which is just a convenient <code>newtype</code> wrapper around <code>(e -> a)</code>, along with an appropriate <code>Monad</code> instance and some <code>Reader</code>-specific utility functions such as <code>ask</code> (retrieve the environment), <code>asks</code> (retrieve a function of the environment), and <code>local</code> (run a subcomputation under a different environment).<br />
</li><br />
<br />
<li>The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Writer-Lazy.html <code>Control.Monad.Writer</code>] module provides the <code>Writer</code> monad, which allows information to be collected as a computation progresses. <code>Writer w a</code> is isomorphic to <code>(a,w)</code>, where the output value <code>a</code> is carried along with an annotation or “log” of type <code>w</code>, which must be an instance of <code>Monoid</code> (see [[#Monoid|section Monoid]]); the special function <code>tell</code> performs logging.<br />
</li><br />
<br />
<li>The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-State-Lazy.html <code>Control.Monad.State</code>] module provides the <code>State s a</code> type, a <code>newtype</code> wrapper around <code>s -> (a,s)</code>. Something of type <code>State s a</code> represents a stateful computation which produces an <code>a</code> but can access and modify the state of type <code>s</code> along the way. The module also provides <code>State</code>-specific utility functions such as <code>get</code> (read the current state), <code>gets</code> (read a function of the current state), <code>put</code> (overwrite the state), and <code>modify</code> (apply a function to the state).<br />
</li><br />
<br />
<li>The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Cont.html <code>Control.Monad.Cont</code>] module provides the <code>Cont</code> monad, which represents computations in continuation-passing style. It can be used to suspend and resume computations, and to implement non-local transfers of control, co-routines, other complex control structures—all in a functionally pure way. <code>Cont</code> has been called the [http://blog.sigfpe.com/2008/12/mother-of-all-monads.html “mother of all monads”] because of its universal properties.<br />
</li><br />
</ul><br />
<br />
{{Exercises|<br />
<ol><br />
<li>Implement a <code>Monad</code> instance for the list constructor, <code>[]</code>. Follow the types!</li><br />
<li>Implement a <code>Monad</code> instance for <code>((->) e)</code>.</li><br />
<li>Implement <code>Functor</code> and <code>Monad</code> instances for <code>Free f</code>, defined as<br />
<haskell><br />
data Free f a = Var a<br />
| Node (f (Free f a))<br />
</haskell><br />
You may assume that <code>f</code> has a <code>Functor</code> instance. This is known as the ''free monad'' built from the functor <code>f</code>.<br />
</li><br />
</ol><br />
}}<br />
<br />
==Intuition==<br />
<br />
Let’s look more closely at the type of <code>(>>=)</code>. The basic intuition is that it combines two computations into one larger computation. The first argument, <code>m a</code>, is the first computation. However, it would be boring if the second argument were just an <code>m b</code>; then there would be no way for the computations to interact with one another (actually, this is exactly the situation with <code>Applicative</code>). So, the second argument to <code>(>>=)</code> has type <code>a -> m b</code>: a function of this type, given a ''result'' of the first computation, can produce a second computation to be run. In other words, <code>x >>= k</code> is a computation which runs <code>x</code>, and then uses the result(s) of <code>x</code> to ''decide'' what computation to run second, using the output of the second computation as the result of the entire computation.<br />
<br />
{{note|Actually, because Haskell allows general recursion, this is a lie: using a Haskell parsing library one can recursively construct ''infinite'' grammars, and hence <code>Applicative</code> (together with <code>Alternative</code>) is enough to parse any context-sensitive language with a finite alphabet. See [http://byorgey.wordpress.com/2012/01/05/parsing-context-sensitive-languages-with-applicative/ Parsing context-sensitive languages with Applicative].}}<br />
Intuitively, it is this ability to use the output from previous computations to decide what computations to run next that makes <code>Monad</code> more powerful than <code>Applicative</code>. The structure of an <code>Applicative</code> computation is fixed, whereas the structure of a <code>Monad</code> computation can change based on intermediate results. This also means that parsers built using an <code>Applicative</code> interface can only parse context-free languages; in order to parse context-sensitive languages a <code>Monad</code> interface is needed.{{noteref}}<br />
<br />
To see the increased power of <code>Monad</code> from a different point of view, let’s see what happens if we try to implement <code>(>>=)</code> in terms of <code>fmap</code>, <code>pure</code>, and <code>(<*>)</code>. We are given a value <code>x</code> of type <code>m a</code>, and a function <code>k</code> of type <code>a -> m b</code>, so the only thing we can do is apply <code>k</code> to <code>x</code>. We can’t apply it directly, of course; we have to use <code>fmap</code> to lift it over the <code>m</code>. But what is the type of <code>fmap k</code>? Well, it’s <code>m a -> m (m b)</code>. So after we apply it to <code>x</code>, we are left with something of type <code>m (m b)</code>—but now we are stuck; what we really want is an <code>m b</code>, but there’s no way to get there from here. We can ''add'' <code>m</code>’s using <code>pure</code>, but we have no way to ''collapse'' multiple <code>m</code>’s into one.<br />
<br />
{{note|1=You might hear some people claim that that the definition in terms of <code>return</code>, <code>fmap</code>, and <code>join</code> is the “math definition” and the definition in terms of <code>return</code> and <code>(>>=)</code> is something specific to Haskell. In fact, both definitions were known in the mathematics community long before Haskell picked up monads.}}<br />
<br />
This ability to collapse multiple <code>m</code>’s is exactly the ability provided by the function <code>join :: m (m a) -> m a</code>, and it should come as no surprise that an alternative definition of <code>Monad</code> can be given in terms of <code>join</code>:<br />
<br />
<haskell><br />
class Applicative m => Monad'' m where<br />
join :: m (m a) -> m a<br />
</haskell><br />
<br />
In fact, the canonical definition of monads in category theory is in terms of <code>return</code>, <code>fmap</code>, and <code>join</code> (often called <math>\eta</math>, <math>T</math>, and <math>\mu</math> in the mathematical literature). Haskell uses an alternative formulation with <code>(>>=)</code> instead of <code>join</code> since it is more convenient to use {{noteref}}. However, sometimes it can be easier to think about <code>Monad</code> instances in terms of <code>join</code>, since it is a more “atomic” operation. (For example, <code>join</code> for the list monad is just <code>concat</code>.)<br />
<br />
{{Exercises|<br />
# Implement <code>(>>{{=}})</code> in terms of <code>fmap</code> (or <code>liftM</code>) and <code>join</code>.<br />
# Now implement <code>join</code> and <code>fmap</code> (<code>liftM</code>) in terms of <code>(>>{{=}})</code> and <code>return</code>.<br />
}}<br />
<br />
==Utility functions==<br />
<br />
The [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad.html <code>Control.Monad</code>] module provides a large number of convenient utility functions, all of which can be implemented in terms of the basic <code>Monad</code> operations (<code>return</code> and <code>(>>=)</code> in particular). We have already seen one of them, namely, <code>join</code>. We also mention some other noteworthy ones here; implementing these utility functions oneself is a good exercise. For a more detailed guide to these functions, with commentary and example code, see Henk-Jan van Tuyl’s [http://members.chello.nl/hjgtuyl/tourdemonad.html tour].<br />
<br />
{{note|Still, it is unclear how this "bug" should be fixed. Making <code>Monad</code> require a <code>Functor</code> instance has some drawbacks, as mentioned in this [http://www.haskell.org/pipermail/haskell-prime/2011-January/003312.html 2011 mailing-list discussion]. —Geheimdienst}}<br />
<br />
* <code>liftM :: Monad m => (a -> b) -> m a -> m b</code>. This should be familiar; of course, it is just <code>fmap</code>. The fact that we have both <code>fmap</code> and <code>liftM</code> is an unfortunate consequence of the fact that the <code>Monad</code> type class does not require a <code>Functor</code> instance, even though mathematically speaking, every monad is a functor. However, <code>fmap</code> and <code>liftM</code> are essentially interchangeable, since it is a bug (in a social rather than technical sense) for any type to be an instance of <code>Monad</code> without also being an instance of <code>Functor</code> {{noteref}}.<br />
<br />
* <code>ap :: Monad m => m (a -> b) -> m a -> m b</code> should also be familiar: it is equivalent to <code>(<*>)</code>, justifying the claim that the <code>Monad</code> interface is strictly more powerful than <code>Applicative</code>. We can make any <code>Monad</code> into an instance of <code>Applicative</code> by setting <code>pure = return</code> and <code>(<*>) = ap</code>.<br />
<br />
* <code>sequence :: Monad m => [m a] -> m [a]</code> takes a list of computations and combines them into one computation which collects a list of their results. It is again something of a historical accident that <code>sequence</code> has a <code>Monad</code> constraint, since it can actually be implemented only in terms of <code>Applicative</code>. There is an additional generalization of <code>sequence</code> to structures other than lists, which will be discussed in the [[#Traversable|section on <code>Traversable</code>]].<br />
<br />
* <code>replicateM :: Monad m => Int -> m a -> m [a]</code> is simply a combination of [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#v:replicate <code>replicate</code>] and <code>sequence</code>.<br />
<br />
* <code>when :: Monad m => Bool -> m () -> m ()</code> conditionally executes a computation, evaluating to its second argument if the test is <code>True</code>, and to <code>return ()</code> if the test is <code>False</code>. A collection of other sorts of monadic conditionals can be found in the [http://hackage.haskell.org/package/IfElse <code>IfElse</code> package].<br />
<br />
* <code>mapM :: Monad m => (a -> m b) -> [a] -> m [b]</code> maps its first argument over the second, and <code>sequence</code>s the results. The <code>forM</code> function is just <code>mapM</code> with its arguments reversed; it is called <code>forM</code> since it models generalized <code>for</code> loops: the list <code>[a]</code> provides the loop indices, and the function <code>a -> m b</code> specifies the “body” of the loop for each index.<br />
<br />
* <code>(=<<) :: Monad m => (a -> m b) -> m a -> m b</code> is just <code>(>>=)</code> with its arguments reversed; sometimes this direction is more convenient since it corresponds more closely to function application.<br />
<br />
* <code>(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c</code> is sort of like function composition, but with an extra <code>m</code> on the result type of each function, and the arguments swapped. We’ll have more to say about this operation later. There is also a flipped variant, <code>(<=<)</code>.<br />
<br />
* The <code>guard</code> function is for use with instances of <code>MonadPlus</code>, which is discussed at the end of the [[#Monoid|<code>Monoid</code> section]].<br />
<br />
Many of these functions also have “underscored” variants, such as <code>sequence_</code> and <code>mapM_</code>; these variants throw away the results of the computations passed to them as arguments, using them only for their side effects.<br />
<br />
Other monadic functions which are occasionally useful include <code>filterM</code>, <code>zipWithM</code>, <code>foldM</code>, and <code>forever</code>. <br />
<br />
==Laws==<br />
<br />
There are several laws that instances of <code>Monad</code> should satisfy (see also the [[Monad laws]] wiki page). The standard presentation is:<br />
<br />
<haskell><br />
return a >>= k = k a<br />
m >>= return = m<br />
m >>= (\x -> k x >>= h) = (m >>= k) >>= h<br />
<br />
fmap f xs = xs >>= return . f = liftM f xs<br />
</haskell><br />
<br />
The first and second laws express the fact that <code>return</code> behaves nicely: if we inject a value <code>a</code> into a monadic context with <code>return</code>, and then bind to <code>k</code>, it is the same as just applying <code>k</code> to <code>a</code> in the first place; if we bind a computation <code>m</code> to <code>return</code>, nothing changes. The third law essentially says that <code>(>>=)</code> is associative, sort of. The last law ensures that <code>fmap</code> and <code>liftM</code> are the same for types which are instances of both <code>Functor</code> and <code>Monad</code>—which, as already noted, should be every instance of <code>Monad</code>.<br />
<br />
{{note|I like to pronounce this operator “fish”.}}<br />
<br />
However, the presentation of the above laws, especially the third, is marred by the asymmetry of <code>(>>=)</code>. It’s hard to look at the laws and see what they’re really saying. I prefer a much more elegant version of the laws, which is formulated in terms of <code>(>=>)</code> {{noteref}}. Recall that <code>(>=>)</code> “composes” two functions of type <code>a -> m b</code> and <code>b -> m c</code>. You can think of something of type <code>a -> m b</code> (roughly) as a function from <code>a</code> to <code>b</code> which may also have some sort of effect in the context corresponding to <code>m</code>. <code>(>=>)</code> lets us compose these “effectful functions”, and we would like to know what properties <code>(>=>)</code> has. The monad laws reformulated in terms of <code>(>=>)</code> are:<br />
<br />
<haskell><br />
return >=> g = g<br />
g >=> return = g<br />
(g >=> h) >=> k = g >=> (h >=> k)<br />
</haskell><br />
<br />
{{note|As fans of category theory will note, these laws say precisely that functions of type <code>a -> m b</code> are the arrows of a category with <code>(>{{=}}>)</code> as composition! Indeed, this is known as the ''Kleisli category'' of the monad <code>m</code>. It will come up again when we discuss <code>Arrow</code>s.}}<br />
<br />
Ah, much better! The laws simply state that <code>return</code> is the identity of <code>(>=>)</code>, and that <code>(>=>)</code> is associative {{noteref}}.<br />
<br />
There is also a formulation of the monad laws in terms of <code>fmap</code>, <code>return</code>, and <code>join</code>; for a discussion of this formulation, see the Haskell [http://en.wikibooks.org/wiki/Haskell/Category_theory wikibook page on category theory].<br />
<br />
{{Exercises|<br />
# Given the definition <code>g >{{=}}> h {{=}} \x -> g x >>{{=}} h</code>, prove the equivalence of the above laws and the usual monad laws.<br />
}}<br />
<br />
==<code>do</code> notation==<br />
<br />
Haskell’s special <code>do</code> notation supports an “imperative style” of programming by providing syntactic sugar for chains of monadic expressions. The genesis of the notation lies in realizing that something like <code>a >>= \x -> b >> c >>= \y -> d </code> can be more readably written by putting successive computations on separate lines:<br />
<br />
<haskell><br />
a >>= \x -><br />
b >><br />
c >>= \y -><br />
d<br />
</haskell><br />
<br />
This emphasizes that the overall computation consists of four computations <code>a</code>, <code>b</code>, <code>c</code>, and <code>d</code>, and that <code>x</code> is bound to the result of <code>a</code>, and <code>y</code> is bound to the result of <code>c</code> (<code>b</code>, <code>c</code>, and <code>d</code> are allowed to refer to <code>x</code>, and <code>d</code> is allowed to refer to <code>y</code> as well). From here it is not hard to imagine a nicer notation:<br />
<br />
<haskell><br />
do { x <- a<br />
; b<br />
; y <- c<br />
; d<br />
}<br />
</haskell><br />
<br />
(The curly braces and semicolons may optionally be omitted; the Haskell parser uses layout to determine where they should be inserted.) This discussion should make clear that <code>do</code> notation is just syntactic sugar. In fact, <code>do</code> blocks are recursively translated into monad operations (almost) like this:<br />
<br />
<pre><br />
do e → e<br />
do { e; stmts } → e >> do { stmts }<br />
do { v <- e; stmts } → e >>= \v -> do { stmts }<br />
do { let decls; stmts} → let decls in do { stmts }<br />
</pre><br />
<br />
This is not quite the whole story, since <code>v</code> might be a pattern instead of a variable. For example, one can write<br />
<br />
<haskell><br />
do (x:xs) <- foo<br />
bar x<br />
</haskell><br />
<br />
but what happens if <code>foo</code> produces an empty list? Well, remember that ugly <code>fail</code> function in the <code>Monad</code> type class declaration? That’s what happens. See [http://haskell.org/onlinereport/exps.html#sect3.14 section 3.14 of the Haskell Report] for the full details. See also the discussion of <code>MonadPlus</code> and <code>MonadZero</code> in the [[#Other monoidal classes: Alternative, MonadPlus, ArrowPlus|section on other monoidal classes]].<br />
<br />
A final note on intuition: <code>do</code> notation plays very strongly to the “computational context” point of view rather than the “container” point of view, since the binding notation <code>x <- m</code> is suggestive of “extracting” a single <code>x</code> from <code>m</code> and doing something with it. But <code>m</code> may represent some sort of a container, such as a list or a tree; the meaning of <code>x <- m</code> is entirely dependent on the implementation of <code>(>>=)</code>. For example, if <code>m</code> is a list, <code>x <- m</code> actually means that <code>x</code> will take on each value from the list in turn.<br />
<br />
==Further reading==<br />
<br />
Philip Wadler was the first to propose using monads to structure functional programs. [http://homepages.inf.ed.ac.uk/wadler/topics/monads.html His paper] is still a readable introduction to the subject.<br />
<br />
{{note|1=<br />
[[All About Monads]],<br />
[http://haskell.org/haskellwiki/Monads_as_Containers Monads as containers],<br />
[http://en.wikibooks.org/w/index.php?title=Haskell/Understanding_monads&oldid=933545 Understanding monads],<br />
[[The Monadic Way]],<br />
[http://blog.sigfpe.com/2006/08/you-could-have-invented-monads-and.html You Could Have Invented Monads! (And Maybe You Already Have.)],<br />
[http://www.haskell.org/pipermail/haskell-cafe/2006-November/019190.html there’s a monster in my Haskell!],<br />
[http://kawagner.blogspot.com/2007/02/understanding-monads-for-real.html Understanding Monads. For real.],<br />
[http://www.randomhacks.net/articles/2007/03/12/monads-in-15-minutes Monads in 15 minutes: Backtracking and Maybe],<br />
[http://haskell.org/haskellwiki/Monads_as_computation Monads as computation],<br />
[http://metafoo.co.uk/practical-monads.txt Practical Monads]}}<br />
<br />
There are, of course, numerous monad tutorials of varying quality {{noteref}}.<br />
<br />
A few of the best include Cale Gibbard’s [http://haskell.org/haskellwiki/Monads_as_Containers Monads as containers] and [http://haskell.org/haskellwiki/Monads_as_computation Monads as computation]; Jeff Newbern’s [[All About Monads]], a comprehensive guide with lots of examples; and Dan Piponi’s [http://blog.sigfpe.com/2006/08/you-could-have-invented-monads-and.html You Could Have Invented Monads!], which features great exercises. If you just want to know how to use <code>IO</code>, you could consult the [[Introduction to IO]]. Even this is just a sampling; the [[monad tutorials timeline]] is a more complete list. (All these monad tutorials have prompted parodies like [http://koweycode.blogspot.com/2007/01/think-of-monad.html think of a monad ...] as well as other kinds of backlash like [http://ahamsandwich.wordpress.com/2007/07/26/monads-and-why-monad-tutorials-are-all-awful/ Monads! (and Why Monad Tutorials Are All Awful)] or [http://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/ Abstraction, intuition, and the “monad tutorial fallacy”].)<br />
<br />
Other good monad references which are not necessarily tutorials include [http://members.chello.nl/hjgtuyl/tourdemonad.html Henk-Jan van Tuyl’s tour] of the functions in <code>Control.Monad</code>, Dan Piponi’s [http://blog.sigfpe.com/2006/10/monads-field-guide.html field guide], Tim Newsham’s [http://www.thenewsh.com/~newsham/haskell/monad.html What’s a Monad?], and Chris Smith's excellent article [http://cdsmith.wordpress.com/2012/04/18/why-do-monads-matter/ Why Do Monads Matter?]. There are also many blog posts which have been written on various aspects of monads; a collection of links can be found under [[Blog articles/Monads]].<br />
<br />
For help constructing monads from scratch, and for obtaining a "deep embedding" of monad operations suitable for use in, say, compiling a domain-specific language, see [http://projects.haskell.org/operational apfelmus's operational package].<br />
<br />
One of the quirks of the <code>Monad</code> class and the Haskell type system is that it is not possible to straightforwardly declare <code>Monad</code> instances for types which require a class constraint on their data, even if they are monads from a mathematical point of view. For example, <code>Data.Set</code> requires an <code>Ord</code> constraint on its data, so it cannot be easily made an instance of <code>Monad</code>. A solution to this problem was [http://www.randomhacks.net/articles/2007/03/15/data-set-monad-haskell-macros first described by Eric Kidd], and later made into a [http://hackage.haskell.org/cgi-bin/hackage-scripts/package/rmonad library named rmonad] by Ganesh Sittampalam and Peter Gavin.<br />
<br />
There are many good reasons for eschewing <code>do</code> notation; some have gone so far as to [[Do_notation_considered_harmful|consider it harmful]].<br />
<br />
Monads can be generalized in various ways; for an exposition of one possibility, see Robert Atkey’s paper on [http://homepages.inf.ed.ac.uk/ratkey/paramnotions-jfp.pdf parameterized monads], or Dan Piponi’s [http://blog.sigfpe.com/2009/02/beyond-monads.html Beyond Monads].<br />
<br />
For the categorically inclined, monads can be viewed as monoids ([http://blog.sigfpe.com/2008/11/from-monoids-to-monads.html From Monoids to Monads]) and also as closure operators [http://blog.plover.com/math/monad-closure.html Triples and Closure]. Derek Elkins’s article in [http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf issue 13 of the Monad.Reader] contains an exposition of the category-theoretic underpinnings of some of the standard <code>Monad</code> instances, such as <code>State</code> and <code>Cont</code>. Jonathan Hill and Keith Clarke have [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.6497 an early paper explaining the connection between monads as they arise in category theory and as used in functional programming]. There is also a [http://okmij.org/ftp/Computation/IO-monad-history.html web page by Oleg Kiselyov] explaining the history of the IO monad.<br />
<br />
Links to many more research papers related to monads can be found under [[Research papers/Monads and arrows]].<br />
<br />
=Monad transformers=<br />
<br />
One would often like to be able to combine two monads into one: for example, to have stateful, nondeterministic computations (<code>State</code> + <code>[]</code>), or computations which may fail and can consult a read-only environment (<code>Maybe</code> + <code>Reader</code>), and so on. Unfortunately, monads do not compose as nicely as applicative functors (yet another reason to use <code>Applicative</code> if you don’t need the full power that <code>Monad</code> provides), but some monads can be combined in certain ways.<br />
<br />
==Standard monad transformers==<br />
<br />
The [http://hackage.haskell.org/package/transformers transformers] library provides a number of standard ''monad transformers''. Each monad transformer adds a particular capability/feature/effect to any existing monad.<br />
<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Identity.html <code>IdentityT</code>] is the identity transformer, which maps a monad to (something isomorphic to) itself. This may seem useless at first glance, but it is useful for the same reason that the <code>id</code> function is useful -- it can be passed as an argument to things which are parameterized over an arbitrary monad transformer, when you do not actually want any extra capabilities.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-State.html <code>StateT</code>] adds a read-write state.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Reader.html <code>ReaderT</code>] adds a read-only environment.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Writer.html <code>WriterT</code>] adds a write-only log.<br />
* [http://hackage.haskell.org/packages/archive/transformers/0.2.2.0/doc/html/Control-Monad-Trans-RWS.html <code>RWST</code>] conveniently combines <code>ReaderT</code>, <code>WriterT</code>, and <code>StateT</code> into one.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Maybe.html <code>MaybeT</code>] adds the possibility of failure.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Error.html <code>ErrorT</code>] adds the possibility of failure with an arbitrary type to represent errors.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-List.html <code>ListT</code>] adds non-determinism (however, see the discussion of <code>ListT</code> below).<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Cont.html <code>ContT</code>] adds continuation handling.<br />
<br />
For example, <code>StateT s Maybe</code> is an instance of <code>Monad</code>; computations of type <code>StateT s Maybe a</code> may fail, and have access to a mutable state of type <code>s</code>. Monad transformers can be multiply stacked. One thing to keep in mind while using monad transformers is that the order of composition matters. For example, when a <code>StateT s Maybe a</code> computation fails, the state ceases being updated (indeed, it simply disappears); on the other hand, the state of a <code>MaybeT (State s) a</code> computation may continue to be modified even after the computation has "failed". This may seem backwards, but it is correct. Monad transformers build composite monads “inside out”; <code>MaybeT (State s) a</code> is isomorphic to <code>s -> (Maybe a, s)</code>. (Lambdabot has an indispensable <code>@unmtl</code> command which you can use to “unpack” a monad transformer stack in this way.)<br />
Intuitively, the monads become "more fundamental" the further down in the stack you get, and the effects of a given monad "have precedence" over the effects of monads further up the stack. Of course, this is just handwaving, and if you are unsure of the proper order for some monads you wish to combine, there is no substitute for using <code>@unmtl</code> or simply trying out the various options.<br />
<br />
==Definition and laws==<br />
<br />
All monad transformers should implement the <code>MonadTrans</code> type class, defined in <code>Control.Monad.Trans.Class</code>:<br />
<br />
<haskell><br />
class MonadTrans t where<br />
lift :: Monad m => m a -> t m a<br />
</haskell><br />
<br />
It allows arbitrary computations in the base monad <code>m</code> to be “lifted” into computations in the transformed monad <code>t m</code>. (Note that type application associates to the left, just like function application, so <code>t m a = (t m) a</code>.)<br />
<br />
<code>lift</code> must satisfy the laws<br />
<haskell><br />
lift . return = return<br />
lift (m >>= f) = lift m >>= (lift . f)<br />
</haskell><br />
which intuitively state that <code>lift</code> transforms <code>m a</code> computations into <code>t m a</code> computations in a "sensible" way, which sends the <code>return</code> and <code>(>>=)</code> of <code>m</code> to the <code>return</code> and <code>(>>=)</code> of <code>t m</code>.<br />
<br />
{{Exercises|<br />
# What is the kind of <code>t</code> in the declaration of <code>MonadTrans</code>?<br />
}}<br />
<br />
==Transformer type classes and "capability" style==<br />
<br />
{{note|The only problem with this scheme is the quadratic number of instances required as the number of standard monad transformers grows—but as the current set of standard monad transformers seems adequate for most common use cases, this may not be that big of a deal.}}<br />
<br />
There are also type classes (provided by the [http://hackage.haskell.org/package/mtl <code>mtl</code> package]) for the operations of each transformer. For example, the <code>MonadState</code> type class provides the state-specific methods <code>get</code> and <code>put</code>, allowing you to conveniently use these methods not only with <code>State</code>, but with any monad which is an instance of <code>MonadState</code>—including <code>MaybeT (State s)</code>, <code>StateT s (ReaderT r IO)</code>, and so on. Similar type classes exist for <code>Reader</code>, <code>Writer</code>, <code>Cont</code>, <code>IO</code>, and others {{noteref}}.<br />
<br />
These type classes serve two purposes. First, they get rid of (most of) the need for explicitly using <code>lift</code>, giving a type-directed way to automatically determine the right number of calls to <code>lift</code>. Simply writing <code>put</code> will be automatically translated into <code>lift . put</code>, <code>lift . lift . put</code>, or something similar depending on what concrete monad stack you are using.<br />
<br />
Second, they give you more flexibility to switch between different concrete monad stacks. For example, if you are writing a state-based algorithm, don't write<br />
<haskell><br />
foo :: State Int Char<br />
foo = modify (*2) >> return 'x'<br />
</haskell><br />
but rather<br />
<haskell><br />
foo :: MonadState Int m => m Char<br />
foo = modify (*2) >> return 'x'<br />
</haskell><br />
Now, if somewhere down the line you realize you need to introduce the possibility of failure, you might switch from <code>State Int</code> to <code>MaybeT (State Int)</code>. The type of the first version of <code>foo</code> would need to be modified to reflect this change, but the second version of <code>foo</code> can still be used as-is.<br />
<br />
However, this sort of "capability-based" style (<i>e.g.</i> specifying that <code>foo</code> works for any monad with the "state capability") quickly runs into problems when you try to naively scale it up: for example, what if you need to maintain two independent states? A framework for solving this and related problems is described by Schrijvers and Olivera ([http://users.ugent.be/~tschrijv/Research/papers/icfp2011.pdf Monads, zippers and views: virtualizing the monad stack, ICFP 2011]) and is implemented in the [http://hackage.haskell.org/package/Monatron <code>Monatron</code> package].<br />
<br />
==Composing monads==<br />
<br />
Is the composition of two monads always a monad? As hinted previously, the answer is no. For example, ''XXX insert example here''.<br />
<br />
Since <code>Applicative</code> functors are closed under composition, the problem must lie with <code>join</code>. Indeed, suppose <code>m</code> and <code>n</code> are arbitrary monads; to make a monad out of their composition we would need to be able to implement<br />
<haskell><br />
join :: m (n (m (n a))) -> m (n a)<br />
</haskell><br />
but it is not clear how this could be done in general. The <code>join</code> method for <code>m</code> is no help, because the two occurrences of <code>m</code> are not next to each other (and likewise for <code>n</code>).<br />
<br />
However, one situation in which it can be done is if <code>n</code> ''distributes'' over <code>m</code>, that is, if there is a function<br />
<haskell><br />
distrib :: n (m a) -> m (n a)<br />
</haskell><br />
satisfying certain laws. See Jones and Duponcheel ([http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.2605 Composing Monads]); see also the [[#Traversable|section on Traversable]].<br />
<br />
{{Exercises|<br />
* Implement <code>join :: M (N (M (N a))) -> M (N a)</code>, given <code>distrib :: N (M a) -> M (N a)</code> and assuming <code>M</code> and <code>N</code> are instances of <code>Monad</code>.<br />
}}<br />
<br />
==Further reading==<br />
<br />
Much of the monad transformer library (originally [http://hackage.haskell.org/package/mtl <code>mtl</code>], now split between <code>mtl</code> and [http://hackage.haskell.org/package/transformers <code>transformers</code>]), including the <code>Reader</code>, <code>Writer</code>, <code>State</code>, and other monads, as well as the monad transformer framework itself, was inspired by Mark Jones’s classic paper [http://web.cecs.pdx.edu/~mpj/pubs/springschool.html Functional Programming with Overloading and Higher-Order Polymorphism]. It’s still very much worth a read—and highly readable—after almost fifteen years.<br />
<br />
See [http://article.gmane.org/gmane.comp.lang.haskell.libraries/17139 Edward Kmett's mailing list message] for a description of the history and relationships among monad transformer packages (<code>mtl</code>, <code>transformers</code>, <code>monads-fd</code>, <code>monads-tf</code>).<br />
<br />
There are two excellent references on monad transformers. Martin Grabmüller’s [http://www.grabmueller.de/martin/www/pub/Transformers.en.html Monad Transformers Step by Step] is a thorough description, with running examples, of how to use monad transformers to elegantly build up computations with various effects. [http://cale.yi.org/index.php/How_To_Use_Monad_Transformers Cale Gibbard’s article] on how to use monad transformers is more practical, describing how to structure code using monad transformers to make writing it as painless as possible. Another good starting place for learning about monad transformers is a [http://blog.sigfpe.com/2006/05/grok-haskell-monad-transformers.html blog post by Dan Piponi].<br />
<br />
The <code>ListT</code> transformer from the <code>transformers</code> package comes with the caveat that <code>ListT m</code> is only a monad when <code>m</code> is ''commutative'', that is, when <code>ma >>= \a -> mb >>= \b -> foo</code> is equivalent to <code>mb >>= \b -> ma >>= \a -> foo</code> (i.e. the order of <code>m</code>'s effects does not matter). For one explanation why, see Dan Piponi's blog post [http://blog.sigfpe.com/2006/11/why-isnt-listt-monad.html "Why isn't <code><nowiki>ListT []</nowiki></code> a monad"]. For more examples, as well as a design for a version of <code>ListT</code> which does not have this problem, see [http://haskell.org/haskellwiki/ListT_done_right <code>ListT</code> done right].<br />
<br />
There is an alternative way to compose monads, using coproducts, as described by [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.3581 Lüth and Ghani]. This method is interesting but has not (yet?) seen widespread use.<br />
<br />
=MonadFix=<br />
<br />
''Note: <code>MonadFix</code> is included here for completeness (and because it is interesting) but seems not to be used much. Skipping this section on a first read-through is perfectly OK (and perhaps even recommended).''<br />
<br />
==<code>mdo</code>/<code>do rec</code> notation==<br />
<br />
{{note|In GHC 7.6, the flag has been changed to <code>-XRecursiveDo</code>.}}<br />
The <code>MonadFix</code> class describes monads which support the special fixpoint operation <code>mfix :: (a -> m a) -> m a</code>, which allows the output of monadic computations to be defined via (effectful) recursion. This is [http://www.haskell.org/ghc/docs/latest/html/users_guide/syntax-extns.html#recursive-do-notation supported in GHC] by a special “recursive do” notation, enabled by the <code>-XDoRec</code> flag{{noteref}}. Within a <code>do</code> block, one may have a nested <code>rec</code> block, like so:<br />
<haskell><br />
do { x <- foo<br />
; rec { y <- baz<br />
; z <- bar<br />
; bob<br />
}<br />
; w <- frob<br />
}<br />
</haskell><br />
Normally (if we had <code>do</code> in place of <code>rec</code> in the above example), <code>y</code> would be in scope in <code>bar</code> and <code>bob</code> but not in <code>baz</code>, and <code>z</code> would be in scope only in <code>bob</code>. With the <code>rec</code>, however, <code>y</code> and <code>z</code> are both in scope in all three of <code>baz</code>, <code>bar</code>, and <code>bob</code>. A <code>rec</code> block is analogous to a <code>let</code> block such as<br />
<haskell><br />
let { y = baz<br />
; z = bar<br />
}<br />
in bob<br />
</haskell><br />
because, in Haskell, every variable bound in a <code>let</code>-block is in scope throughout the entire block. (From this point of view, Haskell's normal <code>do</code> blocks are analogous to Scheme's <code>let*</code> construct.)<br />
<br />
What could such a feature be used for? One of the motivating examples given in the original paper describing <code>MonadFix</code> (see below) is encoding circuit descriptions. A line in a <code>do</code>-block such as <br />
<haskell><br />
x <- gate y z<br />
</haskell><br />
describes a gate whose input wires are labeled <code>y</code> and <code>z</code> and whose output wire is labeled <code>x</code>. Many (most?) useful circuits, however, involve some sort of feedback loop, making them impossible to write in a normal <code>do</code>-block (since some wire would have to be mentioned as an input ''before'' being listed as an output). Using a <code>rec</code> block solves this problem.<br />
<br />
==Examples and intuition==<br />
<br />
Of course, not every monad supports such recursive binding. However, as mentioned above, it suffices to have an implementation of <code>mfix :: (a -> m a) -> m a</code>, satisfying a few laws. Let's try implementing <code>mfix</code> for the <code>Maybe</code> monad. That is, we want to implement a function<br />
<haskell><br />
maybeFix :: (a -> Maybe a) -> Maybe a<br />
</haskell><br />
{{note|Actually, <code>fix</code> is implemented slightly differently for efficiency reasons; but the given definition is equivalent and simpler for the present purpose.}}<br />
Let's think for a moment about the implementation {{noteref}} of the non-monadic <code>fix :: (a -> a) -> a</code>:<br />
<haskell><br />
fix f = f (fix f)<br />
</haskell><br />
Inspired by <code>fix</code>, our first attempt at implementing <code>maybeFix</code> might be something like<br />
<haskell><br />
maybeFix :: (a -> Maybe a) -> Maybe a<br />
maybeFix f = maybeFix f >>= f<br />
</haskell><br />
This has the right type. However, something seems wrong: there is nothing in particular here about <code>Maybe</code>; <code>maybeFix</code> actually has the more general type <code>Monad m => (a -> m a) -> m a</code>. But didn't we just say that not all monads support <code>mfix</code>?<br />
<br />
The answer is that although this implementation of <code>maybeFix</code> has the right type, it does ''not'' have the intended semantics. If we think about how <code>(>>=)</code> works for the <code>Maybe</code> monad (by pattern-matching on its first argument to see whether it is <code>Nothing</code> or <code>Just</code>) we can see that this definition of <code>maybeFix</code> is completely useless: it will just recurse infinitely, trying to decide whether it is going to return <code>Nothing</code> or <code>Just</code>, without ever even so much as a glance in the direction of <code>f</code>.<br />
<br />
The trick is to simply ''assume'' that <code>maybeFix</code> will return <code>Just</code>, and get on with life!<br />
<haskell><br />
maybeFix :: (a -> Maybe a) -> Maybe a<br />
maybeFix f = ma<br />
where ma = f (fromJust ma)<br />
</haskell><br />
This says that the result of <code>maybeFix</code> is <code>ma</code>, and assuming that <code>ma = Just x</code>, it is defined (recursively) to be equal to <code>f x</code>.<br />
<br />
Why is this OK? Isn't <code>fromJust</code> almost as bad as <code>unsafePerformIO</code>? Well, usually, yes. This is just about the only situation in which it is justified! The interesting thing to note is that <code>maybeFix</code> ''will never crash'' -- although it may, of course, fail to terminate. The only way we could get a crash is if we try to evaluate <code>fromJust ma</code> when we know that <code>ma = Nothing</code>. But how could we know <code>ma = Nothing</code>? Since <code>ma</code> is defined as <code>f (fromJust ma)</code>, it must be that this expression has already been evaluated to <code>Nothing</code> -- in which case there is no reason for us to be evaluating <code>fromJust ma</code> in the first place! <br />
<br />
To see this from another point of view, we can consider three possibilities. First, if <code>f</code> outputs <code>Nothing</code> without looking at its argument, then <code>maybeFix f</code> clearly returns <code>Nothing</code>. Second, if <code>f</code> always outputs <code>Just x</code>, where <code>x</code> depends on its argument, then the recursion can proceed usefully: <code>fromJust ma</code> will be able to evaluate to <code>x</code>, thus feeding <code>f</code>'s output back to it as input. Third, if <code>f</code> tries to use its argument to decide whether to output <code>Just</code> or <code>Nothing</code>, then <code>maybeFix f</code> will not terminate: evaluating <code>f</code>'s argument requires evaluating <code>ma</code> to see whether it is <code>Just</code>, which requires evaluating <code>f (fromJust ma)</code>, which requires evaluating <code>ma</code>, ... and so on.<br />
<br />
There are also instances of <code>MonadFix</code> for lists (which works analogously to the instance for <code>Maybe</code>), for <code>ST</code>, and for <code>IO</code>. The [http://hackage.haskell.org/packages/archive/base/latest/doc/html/src/System-IO.html#fixIO instance for <code>IO</code>] is particularly amusing: it creates a new <code>IORef</code> (with a dummy value), immediately reads its contents using <code>unsafeInterleaveIO</code> (which delays the actual reading lazily until the value is needed), uses the contents of the <code>IORef</code> to compute a new value, which it then writes back into the <code>IORef</code>. It almost seems, spookily, that <code>mfix</code> is sending a value back in time to itself through the <code>IORef</code> -- though of course what is really going on is that the reading is delayed just long enough (via <code>unsafeInterleaveIO</code>) to get the process bootstrapped.<br />
<br />
{{Exercises|<br />
* Implement a <code>MonadFix</code> instance for <code>[]</code>.<br />
}}<br />
<br />
==GHC 7.6 changes==<br />
<br />
GHC 7.6 reinstated the old <code>mdo</code> syntax, so the example at the start of this section can be written<br />
<br />
<haskell><br />
mdo { x <- foo<br />
; y <- baz<br />
; z <- bar<br />
; bob<br />
; w <- frob<br />
}<br />
</haskell><br />
<br />
which will be translated into the original example (assuming that, say, <code>bar</code> and <code>bob</code> refer to <code>y</code>. The difference is that <code>mdo</code> will analyze the code in order to find minimal recursive blocks, which will be placed in <code>rec</code> blocks, whereas <code>rec</code> blocks desugar directly into calls to <code>mfix</code> without any further analysis. <br />
==Further reading==<br />
<br />
For more information (such as the precise desugaring rules for <code>rec</code> blocks), see Levent Erkök and John Launchbury's 2002 Haskell workshop paper, [http://sites.google.com/site/leventerkok/recdo.pdf?attredirects=0 A Recursive do for Haskell], or for full details, Levent Erkök’s thesis, [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.15.1543&rep=rep1&type=pdf Value Recursion in Monadic Computations]. (Note, while reading, that <code>MonadFix</code> used to be called <code>MonadRec</code>.) You can also read the [http://www.haskell.org/ghc/docs/latest/html/users_guide/syntax-extns.html#recursive-do-notation GHC user manual section on recursive do-notation].<br />
<br />
=Semigroup=<br />
<br />
A semigroup is a set <math>S\ </math> together with a binary operation <math>\oplus\ </math> which<br />
combines elements from <math>S\ </math>. The <math>\oplus\ </math> operator is required to be associative<br />
(that is, <math>(a \oplus b) \oplus c = a \oplus (b \oplus c)\ </math>, for any<br />
<math>a,b,c\ </math> which are elements of <math>S\ </math>).<br />
<br />
For example, the natural numbers under addition form a semigroup: the sum of any two natural numbers is a natural number, and <math>(a+b)+c = a+(b+c)\ </math> for any natural numbers <math>a\ </math>, <math>b\ </math>, and <math>c\,\ </math>. The integers under multiplication also form a semigroup, as do the integers (or rationals, or reals) under <math>\max\ </math> or <math>\min\ </math>, Boolean values under conjunction and disjunction, lists under concatenation, functions from a set to itself under composition ... Semigroups show up all over the place, once you know to look for them.<br />
<br />
==Definition==<br />
<br />
Semigroups are not (yet?) defined in the base package, but the {{HackagePackage|id=semigroups}} package provides a standard definition.<br />
<br />
The definition of the <code>Semigroup</code> type class ([http://hackage.haskell.org/packages/archive/semigroups/latest/doc/html/Data-Semigroup.html haddock]) is as follows:<br />
<br />
<haskell><br />
class Semigroup a where<br />
(<>) :: a -> a -> a<br />
<br />
sconcat :: NonEmpty a -> a<br />
sconcat = sconcat (a :| as) = go a as where<br />
go b (c:cs) = b <> go c cs<br />
go b [] = b<br />
<br />
times1p :: Whole n => n -> a -> a<br />
times1p = ...<br />
</haskell><br />
<br />
The really important method is <code>(<>)</code>, representing the associative binary operation. The other two methods have default implementations in terms of <code>(<>)</code>, and are included in the type class in case some instances can give more efficient implementations than the default. <code>sconcat</code> reduces a nonempty list using <code>(<>)</code>; <code>times1p n</code> is equivalent to (but more efficient than) <code>sconcat . replicate n</code>. See the [http://hackage.haskell.org/packages/archive/semigroups/latest/doc/html/Data-Semigroup.html haddock documentation] for more information on <code>sconcat</code> and <code>times1p</code>.<br />
<br />
==Laws==<br />
<br />
The only law is that <code>(<>)</code> must be associative:<br />
<br />
<haskell><br />
(x <> y) <> z = x <> (y <> z)<br />
</haskell><br />
<br />
=Monoid=<br />
<br />
Many semigroups have a special element <math>e</math> for which the binary operation <math>\oplus</math> is the identity, that is, <math>e \oplus x = x \oplus e = x</math> for every element <math>x</math>. Such a semigroup-with-identity-element is called a ''monoid''.<br />
<br />
==Definition==<br />
<br />
The definition of the <code>Monoid</code> type class (defined in<br />
<code>Data.Monoid</code>; [http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Monoid.html haddock]) is:<br />
<br />
<haskell><br />
class Monoid a where<br />
mempty :: a<br />
mappend :: a -> a -> a<br />
<br />
mconcat :: [a] -> a<br />
mconcat = foldr mappend mempty<br />
</haskell><br />
<br />
The <code>mempty</code> value specifies the identity element of the monoid, and <code>mappend</code><br />
is the binary operation. The default definition for <code>mconcat</code><br />
“reduces” a list of elements by combining them all with <code>mappend</code>,<br />
using a right fold. It is only in the <code>Monoid</code> class so that specific<br />
instances have the option of providing an alternative, more efficient<br />
implementation; usually, you can safely ignore <code>mconcat</code> when creating<br />
a <code>Monoid</code> instance, since its default definition will work just fine.<br />
<br />
The <code>Monoid</code> methods are rather unfortunately named; they are inspired<br />
by the list instance of <code>Monoid</code>, where indeed <code>mempty = []</code> and <code>mappend = (++)</code>, but this is misleading since many<br />
monoids have little to do with appending (see these [http://thread.gmane.org/gmane.comp.lang.haskell.cafe/50590 Comments from OCaml Hacker Brian Hurt] on the haskell-cafe mailing list).<br />
<br />
==Laws==<br />
<br />
Of course, every <code>Monoid</code> instance should actually be a monoid in the<br />
mathematical sense, which implies these laws:<br />
<br />
<haskell><br />
mempty `mappend` x = x<br />
x `mappend` mempty = x<br />
(x `mappend` y) `mappend` z = x `mappend` (y `mappend` z)<br />
</haskell><br />
<br />
==Instances==<br />
<br />
There are quite a few interesting <code>Monoid</code> instances defined in <code>Data.Monoid</code>.<br />
<br />
<ul><br />
<li><code>[a]</code> is a <code>Monoid</code>, with <code>mempty = []</code> and <code>mappend = (++)</code>. It is not hard to check that <code>(x ++ y) ++ z = x ++ (y ++ z)</code> for any lists <code>x</code>, <code>y</code>, and <code>z</code>, and that the empty list is the identity: <code>[] ++ x = x ++ [] = x</code>.</li><br />
<br />
<li>As noted previously, we can make a monoid out of any numeric type under either addition or multiplication. However, since we can’t have two instances for the same type, <code>Data.Monoid</code> provides two <code>newtype</code> wrappers, <code>Sum</code> and <code>Product</code>, with appropriate <code>Monoid</code> instances.<br />
<br />
<haskell><br />
> getSum (mconcat . map Sum $ [1..5])<br />
15<br />
> getProduct (mconcat . map Product $ [1..5])<br />
120<br />
</haskell><br />
<br />
This example code is silly, of course; we could just write<br />
<code>sum [1..5]</code> and <code>product [1..5]</code>. Nevertheless, these instances are useful in more generalized settings, as we will see in the [[Foldable|section on <code>Foldable</code>]].</li><br />
<br />
<li><code>Any</code> and <code>All</code> are <code>newtype</code> wrappers providing <code>Monoid</code> instances for <code>Bool</code> (under disjunction and conjunction, respectively).</li><br />
<br />
<li> There are three instances for <code>Maybe</code>: a basic instance which lifts a <code>Monoid</code> instance for <code>a</code> to an instance for <code>Maybe a</code>, and two <code>newtype</code> wrappers <code>First</code> and <code>Last</code> for which <code>mappend</code> selects the first (respectively last) non-<code>Nothing</code> item.</li><br />
<br />
<li><code>Endo a</code> is a newtype wrapper for functions <code>a -> a</code>, which form a monoid under composition.</li><br />
<br />
<li>There are several ways to “lift” <code>Monoid</code> instances to instances with additional structure. We have already seen that an instance for <code>a</code> can be lifted to an instance for <code>Maybe a</code>. There are also tuple instances: if <code>a</code> and <code>b</code> are instances of <code>Monoid</code>, then so is <code>(a,b)</code>, using the monoid operations for <code>a</code> and <code>b</code> in the obvious pairwise manner. Finally, if <code>a</code> is a <code>Monoid</code>, then so is the function type <code>e -> a</code> for any <code>e</code>; in particular, <code>g `mappend` h</code> is the function which applies both <code>g</code> and <code>h</code> to its argument and then combines the results using the underlying <code>Monoid</code> instance for <code>a</code>. This can be quite useful and elegant (see [http://thread.gmane.org/gmane.comp.lang.haskell.cafe/52416 example]).</li><br />
<br />
<li>The type <code>Ordering = LT || EQ || GT</code> is a <code>Monoid</code>, defined in such a way that <code>mconcat (zipWith compare xs ys)</code> computes the lexicographic ordering of <code>xs</code> and <code>ys</code> (if <code>xs</code> and <code>ys</code> have the same length). In particular, <code>mempty = EQ</code>, and <code>mappend</code> evaluates to its leftmost non-<code>EQ</code> argument (or <code>EQ</code> if both arguments are <code>EQ</code>). This can be used together with the function instance of <code>Monoid</code> to do some clever things ([http://www.reddit.com/r/programming/comments/7cf4r/monoids_in_my_programming_language/c06adnx example]).</li><br />
<br />
<li>There are also <code>Monoid</code> instances for several standard data structures in the containers library ([http://hackage.haskell.org/packages/archive/containers/0.2.0.0/doc/html/index.html haddock]), including <code>Map</code>, <code>Set</code>, and <code>Sequence</code>.</li><br />
</ul><br />
<br />
<code>Monoid</code> is also used to enable several other type class instances.<br />
As noted previously, we can use <code>Monoid</code> to make <code>((,) e)</code> an instance of <code>Applicative</code>:<br />
<br />
<haskell><br />
instance Monoid e => Applicative ((,) e) where<br />
pure x = (mempty, x)<br />
(u, f) <*> (v, x) = (u `mappend` v, f x)<br />
</haskell><br />
<br />
<code>Monoid</code> can be similarly used to make <code>((,) e)</code> an instance of <code>Monad</code> as well; this is known as the ''writer monad''. As we’ve already seen, <code>Writer</code> and <code>WriterT</code> are a newtype wrapper and transformer for this monad, respectively.<br />
<br />
<code>Monoid</code> also plays a key role in the <code>Foldable</code> type class (see section [[#Foldable|Foldable]]).<br />
<br />
==Other monoidal classes: Alternative, MonadPlus, ArrowPlus==<br />
<br />
The <code>Alternative</code> type class ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html#g:2 haddock])<br />
is for <code>Applicative</code> functors which also have<br />
a monoid structure:<br />
<br />
<haskell><br />
class Applicative f => Alternative f where<br />
empty :: f a<br />
(<|>) :: f a -> f a -> f a<br />
</haskell><br />
<br />
Of course, instances of <code>Alternative</code> should satisfy the monoid laws<br />
<br />
<haskell><br />
empty <|> x = x<br />
x <|> empty = x<br />
(x <|> y) <|> z = x <|> (y <|> z)<br />
</haskell><br />
<br />
Likewise, <code>MonadPlus</code> ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad.html#t:MonadPlus haddock])<br />
is for <code>Monad</code>s with a monoid structure:<br />
<br />
<haskell><br />
class Monad m => MonadPlus m where<br />
mzero :: m a<br />
mplus :: m a -> m a -> m a<br />
</haskell><br />
<br />
The <code>MonadPlus</code> documentation states that it is intended to model<br />
monads which also support “choice and failure”; in addition to the<br />
monoid laws, instances of <code>MonadPlus</code> are expected to satisfy<br />
<br />
<haskell><br />
mzero >>= f = mzero<br />
v >> mzero = mzero<br />
</haskell><br />
<br />
which explains the sense in which <code>mzero</code> denotes failure. Since<br />
<code>mzero</code> should be the identity for <code>mplus</code>, the computation <code>m1 `mplus` m2</code> succeeds (evaluates to something other than <code>mzero</code>) if<br />
either <code>m1</code> or <code>m2</code> does; so <code>mplus</code> represents choice. The <code>guard</code><br />
function can also be used with instances of <code>MonadPlus</code>; it requires a<br />
condition to be satisfied and fails (using <code>mzero</code>) if it is not. A<br />
simple example of a <code>MonadPlus</code> instance is <code>[]</code>, which is exactly the<br />
same as the <code>Monoid</code> instance for <code>[]</code>: the empty list represents<br />
failure, and list concatenation represents choice. In general,<br />
however, a <code>MonadPlus</code> instance for a type need not be the same as its<br />
<code>Monoid</code> instance; <code>Maybe</code> is an example of such a type. A great<br />
introduction to the <code>MonadPlus</code> type class, with interesting examples<br />
of its use, is Doug Auclair’s ''MonadPlus: What a Super Monad!'' in [http://www.haskell.org/wikiupload/6/6a/TMR-Issue11.pdf the Monad.Reader issue 11].<br />
<br />
There used to be a type class called <code>MonadZero</code> containing only<br />
<code>mzero</code>, representing monads with failure. The <code>do</code>-notation requires<br />
some notion of failure to deal with failing pattern matches.<br />
Unfortunately, <code>MonadZero</code> was scrapped in favor of adding the <code>fail</code><br />
method to the <code>Monad</code> class. If we are lucky, someday <code>MonadZero</code> will<br />
be restored, and <code>fail</code> will be banished to the bit bucket where it<br />
belongs (see [[MonadPlus reform proposal]]). The idea is that any<br />
<code>do</code>-block which uses pattern matching (and hence may fail) would require<br />
a <code>MonadZero</code> constraint; otherwise, only a <code>Monad</code> constraint would be<br />
required.<br />
<br />
Finally, <code>ArrowZero</code> and <code>ArrowPlus</code> ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html#t:ArrowZero haddock])<br />
represent <code>Arrow</code>s ([[#Arrow|see below]]) with a<br />
monoid structure:<br />
<br />
<haskell><br />
class Arrow arr => ArrowZero arr where<br />
zeroArrow :: b `arr` c<br />
<br />
class ArrowZero arr => ArrowPlus arr where<br />
(<+>) :: (b `arr` c) -> (b `arr` c) -> (b `arr` c)<br />
</haskell><br />
<br />
==Further reading==<br />
<br />
Monoids have gotten a fair bit of attention recently, ultimately due<br />
to<br />
[http://enfranchisedmind.com/blog/posts/random-thoughts-on-haskell/ a blog post by Brian Hurt], in which he<br />
complained about the fact that the names of many Haskell type classes<br />
(<code>Monoid</code> in particular) are taken from abstract mathematics. This<br />
resulted in [http://thread.gmane.org/gmane.comp.lang.haskell.cafe/50590 a long haskell-cafe thread]<br />
arguing the point and discussing monoids in general.<br />
<br />
{{note|May its name live forever.}}<br />
<br />
However, this was quickly followed by several blog posts about<br />
<code>Monoid</code> {{noteref}}. First, Dan Piponi<br />
wrote a great introductory post, [http://blog.sigfpe.com/2009/01/haskell-monoids-and-their-uses.html Haskell Monoids and their Uses]. This was quickly followed by<br />
Heinrich Apfelmus’s [http://apfelmus.nfshost.com/monoid-fingertree.html Monoids and Finger Trees], an accessible exposition of<br />
Hinze and Paterson’s [http://www.soi.city.ac.uk/%7Eross/papers/FingerTree.html classic paper on 2-3 finger trees], which makes very clever<br />
use of <code>Monoid</code> to implement an elegant and generic data structure.<br />
Dan Piponi then wrote two fascinating articles about using <code>Monoids</code><br />
(and finger trees): [http://blog.sigfpe.com/2009/01/fast-incremental-regular-expression.html Fast Incremental Regular Expressions] and [http://blog.sigfpe.com/2009/01/beyond-regular-expressions-more.html Beyond Regular Expressions]<br />
<br />
In a similar vein, David Place’s article on improving <code>Data.Map</code> in<br />
order to compute incremental folds (see [http://www.haskell.org/sitewiki/images/6/6a/TMR-Issue11.pdf the Monad Reader issue 11])<br />
is also a<br />
good example of using <code>Monoid</code> to generalize a data structure.<br />
<br />
Some other interesting examples of <code>Monoid</code> use include [http://www.reddit.com/r/programming/comments/7cf4r /monoids_in_my_programming_language/c06adnx building elegant list sorting combinators], [http://byorgey.wordpress.com/2008/04/17/collecting-unstructured-information-with-the-monoid-of-partial-knowledge/ collecting unstructured information], [http://izbicki.me/blog/gausian-distributions-are-monoids combining probability distributions], and a brilliant series of posts by Chung-Chieh Shan and Dylan Thurston using <code>Monoid</code>s to [http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers1/ elegantly solve a difficult combinatorial puzzle] (followed by [http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers2/ part 2], [http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers3/ part 3], [http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers4/ part 4]).<br />
<br />
As unlikely as it sounds, monads can actually be viewed as a sort of<br />
monoid, with <code>join</code> playing the role of the binary operation and<br />
<code>return</code> the role of the identity; see [http://blog.sigfpe.com/2008/11/from-monoids-to-monads.html Dan Piponi’s blog post].<br />
<br />
=Foldable=<br />
<br />
The <code>Foldable</code> class, defined in the <code>Data.Foldable</code><br />
module ([http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Foldable.html haddock]), abstracts over containers which can be<br />
“folded” into a summary value. This allows such folding operations<br />
to be written in a container-agnostic way.<br />
<br />
==Definition==<br />
<br />
The definition of the <code>Foldable</code> type class is:<br />
<br />
<haskell><br />
class Foldable t where<br />
fold :: Monoid m => t m -> m<br />
foldMap :: Monoid m => (a -> m) -> t a -> m<br />
<br />
foldr :: (a -> b -> b) -> b -> t a -> b<br />
foldl :: (a -> b -> a) -> a -> t b -> a<br />
foldr1 :: (a -> a -> a) -> t a -> a<br />
foldl1 :: (a -> a -> a) -> t a -> a<br />
</haskell><br />
<br />
This may look complicated, but in fact, to make a <code>Foldable</code> instance<br />
you only need to implement one method: your choice of <code>foldMap</code> or<br />
<code>foldr</code>. All the other methods have default implementations in terms<br />
of these, and are presumably included in the class in case more<br />
efficient implementations can be provided.<br />
<br />
==Instances and examples==<br />
<br />
The type of <code>foldMap</code> should make it clear what it is supposed to do:<br />
given a way to convert the data in a container into a <code>Monoid</code> (a<br />
function <code>a -> m</code>) and a container of <code>a</code>’s (<code>t a</code>), <code>foldMap</code><br />
provides a way to iterate over the entire contents of the container,<br />
converting all the <code>a</code>’s to <code>m</code>’s and combining all the <code>m</code>’s with<br />
<code>mappend</code>. The following code shows two examples: a simple<br />
implementation of <code>foldMap</code> for lists, and a binary tree example<br />
provided by the <code>Foldable</code> documentation.<br />
<br />
<haskell><br />
instance Foldable [] where<br />
foldMap g = mconcat . map g<br />
<br />
data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)<br />
<br />
instance Foldable Tree where<br />
foldMap f Empty = mempty<br />
foldMap f (Leaf x) = f x<br />
foldMap f (Node l k r) = foldMap f l `mappend` f k `mappend` foldMap f r<br />
</haskell><br />
<br />
The <code>foldr</code> function has a type similar to the <code>foldr</code> found in the <code>Prelude</code>, but<br />
more general, since the <code>foldr</code> in the <code>Prelude</code> works only on lists.<br />
<br />
The <code>Foldable</code> module also provides instances for <code>Maybe</code> and <code>Array</code>;<br />
additionally, many of the data structures found in the standard [http://hackage.haskell.org/package/containers containers library] (for example, <code>Map</code>, <code>Set</code>, <code>Tree</code>,<br />
and <code>Sequence</code>) provide their own <code>Foldable</code> instances.<br />
<br />
{{Exercises|<br />
# What is the type of <code>foldMap . foldMap</code>? Or <code>foldMap . foldMap . foldMap</code>, etc.? What do they do?<br />
}}<br />
<br />
==Derived folds==<br />
<br />
Given an instance of <code>Foldable</code>, we can write generic,<br />
container-agnostic functions such as:<br />
<br />
<haskell><br />
-- Compute the size of any container.<br />
containerSize :: Foldable f => f a -> Int<br />
containerSize = getSum . foldMap (const (Sum 1))<br />
<br />
-- Compute a list of elements of a container satisfying a predicate.<br />
filterF :: Foldable f => (a -> Bool) -> f a -> [a]<br />
filterF p = foldMap (\a -> if p a then [a] else [])<br />
<br />
-- Get a list of all the Strings in a container which include the<br />
-- letter a.<br />
aStrings :: Foldable f => f String -> [String]<br />
aStrings = filterF (elem 'a')<br />
</haskell><br />
<br />
The <code>Foldable</code> module also provides a large number of predefined<br />
folds, many of which are generalized versions of <code>Prelude</code> functions of the<br />
same name that only work on lists: <code>concat</code>, <code>concatMap</code>, <code>and</code>,<br />
<code>or</code>, <code>any</code>, <code>all</code>, <code>sum</code>, <code>product</code>, <code>maximum</code>(<code>By</code>),<br />
<code>minimum</code>(<code>By</code>), <code>elem</code>, <code>notElem</code>, and <code>find</code>.<br />
<br />
The important function <code>toList</code> is also provided, which turns any <code>Foldable</code> structure into a list of its elements in left-right order; it works by folding with the list monoid.<br />
<br />
There are also generic functions that work with <code>Applicative</code> or<br />
<code>Monad</code> instances to generate some sort of computation from each<br />
element in a container, and then perform all the side effects from<br />
those computations, discarding the results: <code>traverse_</code>, <code>sequenceA_</code>,<br />
and others. The results must be discarded because the <code>Foldable</code><br />
class is too weak to specify what to do with them: we cannot, in<br />
general, make an arbitrary <code>Applicative</code> or <code>Monad</code> instance into a <code>Monoid</code>, but we can make <code>m ()</code> into a <code>Monoid</code> for any such <code>m</code>. If we do have an <code>Applicative</code> or <code>Monad</code> with a monoid<br />
structure—that is, an <code>Alternative</code> or a <code>MonadPlus</code>—then we can<br />
use the <code>asum</code> or <code>msum</code> functions, which can combine the results as<br />
well. Consult the [http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Foldable.html <code>Foldable</code> documentation] for<br />
more details on any of these functions.<br />
<br />
Note that the <code>Foldable</code> operations always forget the structure of<br />
the container being folded. If we start with a container of type <code>t a</code> for some <code>Foldable t</code>, then <code>t</code> will never appear in the output<br />
type of any operations defined in the <code>Foldable</code> module. Many times<br />
this is exactly what we want, but sometimes we would like to be able<br />
to generically traverse a container while preserving its<br />
structure—and this is exactly what the <code>Traversable</code> class provides,<br />
which will be discussed in the next section.<br />
<br />
{{Exercises|<br />
# Implement <code>toList :: Foldable f {{=}}> f a -> [a]</code>.<br />
# Pick some of the following functions to implement: <code>concat</code>, <code>concatMap</code>, <code>and</code>, <code>or</code>, <code>any</code>, <code>all</code>, <code>sum</code>, <code>product</code>, <code>maximum</code>(<code>By</code>), <code>minimum</code>(<code>By</code>), <code>elem</code>, <code>notElem</code>, and <code>find</code>. Figure out how they generalize to <code>Foldable</code> and come up with elegant implementations using <code>fold</code> or <code>foldMap</code> along with appropriate <code>Monoid</code> instances.<br />
}}<br />
<br />
==Foldable actually isn't==<br />
<br />
The generic term "fold" is often used to refer to the more technical concept of [[Catamorphisms|catamorphism]]. Intuitively, given a way to summarize "one level of structure" (where recursive subterms have already been replaced with their summaries), a catamorphism can summarize an entire recursive structure. It is important to realize that <code>Foldable</code> does <i>not</i> correspond to catamorphisms, but to something weaker. In particular, <code>Foldable</code> allows observing only the left-right order of elements within a structure, not the actual structure itself. Put another way, every use of <code>Foldable</code> can be expressed in terms of <code>toList</code>. For example, <code>fold</code> itself is equivalent to <code>mconcat . toList</code>.<br />
<br />
This is sufficient for many tasks, but not all. For example, consider trying to compute the depth of a <code>Tree</code>: try as we might, there is no way to implement it using <code>Foldable</code>. However, it <i>can</i> be implemented as a catamorphism.<br />
<br />
==Further reading==<br />
<br />
The <code>Foldable</code> class had its genesis in [http://www.soi.city.ac.uk/~ross/papers/Applicative.html McBride and Paterson’s paper]<br />
introducing <code>Applicative</code>, although it has<br />
been fleshed out quite a bit from the form in the paper.<br />
<br />
An interesting use of <code>Foldable</code> (as well as <code>Traversable</code>) can be<br />
found in Janis Voigtländer’s paper [http://doi.acm.org/10.1145/1480881.1480904 Bidirectionalization for free!].<br />
<br />
=Traversable=<br />
<br />
==Definition==<br />
<br />
The <code>Traversable</code> type class, defined in the <code>Data.Traversable</code><br />
module ([http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Traversable.html haddock]), is:<br />
<br />
<haskell><br />
class (Functor t, Foldable t) => Traversable t where<br />
traverse :: Applicative f => (a -> f b) -> t a -> f (t b)<br />
sequenceA :: Applicative f => t (f a) -> f (t a)<br />
mapM :: Monad m => (a -> m b) -> t a -> m (t b)<br />
sequence :: Monad m => t (m a) -> m (t a)<br />
</haskell><br />
<br />
As you can see, every <code>Traversable</code> is also a foldable functor. Like<br />
<code>Foldable</code>, there is a lot in this type class, but making instances is<br />
actually rather easy: one need only implement <code>traverse</code> or<br />
<code>sequenceA</code>; the other methods all have default implementations in<br />
terms of these functions. A good exercise is to figure out what the default<br />
implementations should be: given either <code>traverse</code> or <code>sequenceA</code>, how<br />
would you define the other three methods? (Hint for <code>mapM</code>:<br />
<code>Control.Applicative</code> exports the <code>WrapMonad</code> newtype, which makes any<br />
<code>Monad</code> into an <code>Applicative</code>. The <code>sequence</code> function can be implemented in terms<br />
of <code>mapM</code>.)<br />
<br />
==Intuition==<br />
<br />
The key method of the <code>Traversable</code> class, and the source of its<br />
unique power, is <code>sequenceA</code>. Consider its type:<br />
<haskell><br />
sequenceA :: Applicative f => t (f a) -> f (t a)<br />
</haskell><br />
This answers the fundamental question: when can we commute two<br />
functors? For example, can we turn a tree of lists into a list of<br />
trees?<br />
<br />
The ability to compose two monads depends crucially on this ability to<br />
commute functors. Intuitively, if we want to build a composed monad<br />
<code>M a = m (n a)</code> out of monads <code>m</code> and <code>n</code>, then to be able to<br />
implement <code>join :: M (M a) -> M a</code>, that is,<br />
<code>join :: m (n (m (n a))) -> m (n a)</code>, we have to be able to commute<br />
the <code>n</code> past the <code>m</code> to get <code>m (m (n (n a)))</code>, and then we can use the<br />
<code>join</code>s for <code>m</code> and <code>n</code> to produce something of type <code>m (n a)</code>. See<br />
[http://web.cecs.pdx.edu/~mpj/pubs/springschool.html Mark Jones’s paper] for more details.<br />
<br />
Alternatively, looking at the type of <code>traverse</code>,<br />
<haskell><br />
traverse :: Applicative f => (a -> f b) -> t a -> f (t b)<br />
</haskell><br />
leads us to view <code>Traversable</code> as a generalization of <code>Functor</code>. <code>traverse</code> is an "effectful <code>fmap</code>": it allows us to map over a structure of type <code>t a</code>, applying a function to every element of type <code>a</code> and in order to produce a new structure of type <code>t b</code>; but along the way the function may have some effects (captured by the applicative functor <code>f</code>).<br />
<br />
{{Exercises|<br />
# There are at least two natural ways to turn a tree of lists into a list of trees. What are they, and why?<br />
# Give a natural way to turn a list of trees into a tree of lists.<br />
# What is the type of <code>traverse . traverse</code>? What does it do?<br />
}}<br />
<br />
==Instances and examples==<br />
<br />
What’s an example of a <code>Traversable</code> instance?<br />
The following code shows an example instance for the same<br />
<code>Tree</code> type used as an example in the previous <code>Foldable</code> section. It<br />
is instructive to compare this instance with a <code>Functor</code> instance for<br />
<code>Tree</code>, which is also shown.<br />
<br />
<haskell><br />
data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)<br />
<br />
instance Traversable Tree where<br />
traverse g Empty = pure Empty<br />
traverse g (Leaf x) = Leaf <$> g x<br />
traverse g (Node l x r) = Node <$> traverse g l<br />
<*> g x<br />
<*> traverse g r<br />
<br />
instance Functor Tree where<br />
fmap g Empty = Empty<br />
fmap g (Leaf x) = Leaf $ g x<br />
fmap g (Node l x r) = Node (fmap g l)<br />
(g x)<br />
(fmap g r)<br />
</haskell><br />
<br />
It should be clear that the <code>Traversable</code> and <code>Functor</code> instances for<br />
<code>Tree</code> are almost identical; the only difference is that the <code>Functor</code><br />
instance involves normal function application, whereas the<br />
applications in the <code>Traversable</code> instance take place within an<br />
<code>Applicative</code> context, using <code>(<$>)</code> and <code>(<*>)</code>. In fact, this will<br />
be<br />
true for any type.<br />
<br />
Any <code>Traversable</code> functor is also <code>Foldable</code>, and a <code>Functor</code>. We can see<br />
this not only from the class declaration, but by the fact that we can<br />
implement the methods of both classes given only the <code>Traversable</code><br />
methods.<br />
<br />
The standard libraries provide a number of <code>Traversable</code> instances,<br />
including instances for <code>[]</code>, <code>Maybe</code>, <code>Map</code>, <code>Tree</code>, and <code>Sequence</code>.<br />
Notably, <code>Set</code> is not <code>Traversable</code>, although it is <code>Foldable</code>.<br />
<br />
{{Exercises|<br />
# Implement <code>fmap</code> and <code>foldMap</code> using only the <code>Traversable</code> methods. (Note that the <code>Traversable</code> module provides these implementations as <code>fmapDefault</code> and <code>foldMapDefault</code>.)<br />
}}<br />
<br />
==Laws==<br />
<br />
Any instance of <code>Traversable</code> must statisfy the following two laws, where <code>Identity</code> is the identity functor (as defined in the [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Data-Functor-Identity.html <code>Data.Functor.Identity</code> module] from the <code>transformers</code> package), and <code>Compose</code> wraps the composition of two functors (as defined in [http://hackage.haskell.org/packages/archive/transformers/0.3.0.0/doc/html/Data-Functor-Compose.html <code>Data.Functor.Compose</code>]):<br />
<br />
# <code>traverse Identity = Identity</code><br />
# <code>traverse (Compose . fmap g . f) = Compose . fmap (traverse g) . traverse f</code><br />
<br />
The first law essentially says that traversals cannot make up arbitrary effects. The second law explains how doing two traversals in sequence can be collapsed to a single traversal.<br />
<br />
Additionally, suppose <code>eta</code> is an "<code>Applicative</code> morphism", that is,<br />
<haskell><br />
eta :: forall a f g. (Applicative f, Applicative g) => f a -> g a<br />
</haskell><br />
and <code>eta</code> preserves the <code>Applicative</code> operations: <code>eta (pure x) = pure x</code> and <code>eta (x <*> y) = eta x <*> eta y</code>. Then, by parametricity, any instance of <code>Traversable</code> satisfying the above two laws will also satisfy <code>eta . traverse f = traverse (eta . f)</code>.<br />
<br />
==Further reading==<br />
<br />
The <code>Traversable</code> class also had its genesis in [http://www.soi.city.ac.uk/~ross/papers/Applicative.html McBride and Paterson’s <code>Applicative</code> paper],<br />
and is described in more detail in Gibbons and Oliveira, [http://www.comlab.ox.ac.uk/jeremy.gibbons/publications/iterator.pdf The Essence of the Iterator Pattern],<br />
which also contains a wealth of references to related work.<br />
<br />
<code>Traversable</code> forms a core component of Edward Kmett's [http://hackage.haskell.org/package/lens lens library]. Watching [https://vimeo.com/56063074 Edward's talk on the subject] is a highly recommended way to gain better insight into <code>Traversable</code>, <code>Foldable</code>, <code>Applicative</code>, and many other things besides.<br />
<br />
For references on the <code>Traversable</code> laws, see Russell O'Connor's [http://article.gmane.org/gmane.comp.lang.haskell.libraries/17778 mailing list post] (and subsequent thread).<br />
<br />
=Category=<br />
<br />
<code>Category</code> is a relatively recent addition to the Haskell standard libraries. It generalizes the notion of function composition to general “morphisms”.<br />
<br />
{{note|GHC 7.6.1 changed its rules regarding types and type variables. Now, any operator at the type level is treated as a type ''constructor'' rather than a type ''variable''; prior to GHC 7.6.1 it was possible to use <code>(~&gt;)</code> instead of <code>`arr`</code>. For more information, see [http://thread.gmane.org/gmane.comp.lang.haskell.glasgow.user/21350 the discussion on the GHC-users mailing list]. For a new approach to nice arrow notation that works with GHC 7.6.1, see [http://article.gmane.org/gmane.comp.lang.haskell.glasgow.user/22615 this messsage] and also [http://article.gmane.org/gmane.comp.lang.haskell.glasgow.user/22616 this message] from Edward Kmett, though for simplicity I haven't adopted it here.}}<br />
The definition of the <code>Category</code> type class (from<br />
<code>Control.Category</code>—[http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Category.html haddock]) is shown below. For ease of reading, note that I have used an infix type variable <code>`arr`</code>, in parallel with the infix function type constructor <code>(->)</code>. {{noteref}} This syntax is not part of Haskell 2010. The second definition shown is the one used in the standard libraries. For the remainder of this document, I will use the infix type constructor <code>`arr`</code> for <code>Category</code> as well as <code>Arrow</code>.<br />
<br />
<haskell><br />
class Category arr where<br />
id :: a `arr` a<br />
(.) :: (b `arr` c) -> (a `arr` b) -> (a `arr` c)<br />
<br />
-- The same thing, with a normal (prefix) type constructor<br />
class Category cat where<br />
id :: cat a a<br />
(.) :: cat b c -> cat a b -> cat a c<br />
</haskell><br />
<br />
Note that an instance of <code>Category</code> should be a type constructor which takes two type arguments, that is, something of kind <code>* -> * -> *</code>. It is instructive to imagine the type constructor variable <code>cat</code> replaced by the function constructor <code>(->)</code>: indeed, in this case we recover precisely the familiar identity function <code>id</code> and function composition operator <code>(.)</code> defined in the standard <code>Prelude</code>.<br />
<br />
Of course, the <code>Category</code> module provides exactly such an instance of<br />
<code>Category</code> for <code>(->)</code>. But it also provides one other instance, shown below, which should be familiar from the previous discussion of the <code>Monad</code> laws. <code>Kleisli m a b</code>, as defined in the <code>Control.Arrow</code> module, is just a <code>newtype</code> wrapper around <code>a -> m b</code>.<br />
<br />
<haskell><br />
newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b }<br />
<br />
instance Monad m => Category (Kleisli m) where<br />
id = Kleisli return<br />
Kleisli g . Kleisli h = Kleisli (h >=> g)<br />
</haskell><br />
<br />
The only law that <code>Category</code> instances should satisfy is that <code>id</code> and <code>(.)</code> should form a monoid—that is, <code>id</code> should be the identity of <code>(.)</code>, and <code>(.)</code> should be associative.<br />
<br />
Finally, the <code>Category</code> module exports two additional operators:<br />
<code>(<<<)</code>, which is just a synonym for <code>(.)</code>, and <code>(>>>)</code>, which is <code>(.)</code> with its arguments reversed. (In previous versions of the libraries, these operators were defined as part of the <code>Arrow</code> class.)<br />
<br />
==Further reading==<br />
<br />
The name <code>Category</code> is a bit misleading, since the <code>Category</code> class cannot represent arbitrary categories, but only categories whose objects are objects of <code>Hask</code>, the category of Haskell types. For a more general treatment of categories within Haskell, see the [http://hackage.haskell.org/package/category-extras category-extras package]. For more about category theory in general, see the excellent [http://en.wikibooks.org/wiki/Haskell/Category_theory Haskell wikibook page],<br />
[http://books.google.com/books/about/Category_theory.html?id=-MCJ6x2lC7oC Steve Awodey’s new book], Benjamin Pierce’s [http://books.google.com/books/about/Basic_category_theory_for_computer_scien.html?id=ezdeaHfpYPwC Basic category theory for computer scientists], or [http://folli.loria.fr/cds/1999/esslli99/courses/barr-wells.html Barr and Wells’s category theory lecture notes]. [http://dekudekuplex.wordpress.com/2009/01/19/motivating-learning-category-theory-for-non-mathematicians/ Benjamin Russell’s blog post]<br />
is another good source of motivation and category theory links. You certainly don’t need to know any category theory to be a successful and productive Haskell programmer, but it does lend itself to much deeper appreciation of Haskell’s underlying theory.<br />
<br />
=Arrow=<br />
<br />
The <code>Arrow</code> class represents another abstraction of computation, in a<br />
similar vein to <code>Monad</code> and <code>Applicative</code>. However, unlike <code>Monad</code><br />
and <code>Applicative</code>, whose types only reflect their output, the type of<br />
an <code>Arrow</code> computation reflects both its input and output. Arrows<br />
generalize functions: if <code>arr</code> is an instance of <code>Arrow</code>, a value of<br />
type <code>b `arr` c</code> can be thought of as a computation which takes values of<br />
type <code>b</code> as input, and produces values of type <code>c</code> as output. In the<br />
<code>(->)</code> instance of <code>Arrow</code> this is just a pure function; in general, however,<br />
an arrow may represent some sort of “effectful” computation.<br />
<br />
==Definition==<br />
<br />
The definition of the <code>Arrow</code> type class, from<br />
<code>Control.Arrow</code> ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html haddock]), is:<br />
<br />
<haskell><br />
class Category arr => Arrow arr where<br />
arr :: (b -> c) -> (b `arr` c)<br />
first :: (b `arr` c) -> ((b, d) `arr` (c, d))<br />
second :: (b `arr` c) -> ((d, b) `arr` (d, c))<br />
(***) :: (b `arr` c) -> (b' `arr` c') -> ((b, b') `arr` (c, c'))<br />
(&&&) :: (b `arr` c) -> (b `arr` c') -> (b `arr` (c, c'))<br />
</haskell><br />
<br />
{{note|In versions of the <code>base</code><br />
package prior to version 4, there is no <code>Category</code> class, and the<br />
<code>Arrow</code> class includes the arrow composition operator <code>(>>>)</code>. It<br />
also includes <code>pure</code> as a synonym for <code>arr</code>, but this was removed<br />
since it conflicts with the <code>pure</code> from <code>Applicative</code>.}}<br />
<br />
The first thing to note is the <code>Category</code> class constraint, which<br />
means that we get identity arrows and arrow composition for free:<br />
given two arrows <code>g :: b `arr` c</code> and <code>h :: c `arr` d</code>, we can form their<br />
composition <code>g >>> h :: b `arr` d</code> {{noteref}}.<br />
<br />
As should be a familiar pattern by now, the only methods which must be<br />
defined when writing a new instance of <code>Arrow</code> are <code>arr</code> and <code>first</code>;<br />
the other methods have default definitions in terms of these, but are<br />
included in the <code>Arrow</code> class so that they can be overridden with more<br />
efficient implementations if desired.<br />
<br />
==Intuition==<br />
<br />
Let’s look at each of the arrow methods in turn. [http://www.haskell.org/arrows/ Ross Paterson’s web page on arrows] has nice diagrams which can help<br />
build intuition.<br />
<br />
* The <code>arr</code> function takes any function <code>b -> c</code> and turns it into a generalized arrow <code>b `arr` c</code>. The <code>arr</code> method justifies the claim that arrows generalize functions, since it says that we can treat any function as an arrow. It is intended that the arrow <code>arr g</code> is “pure” in the sense that it only computes <code>g</code> and has no “effects” (whatever that might mean for any particular arrow type).<br />
<br />
* The <code>first</code> method turns any arrow from <code>b</code> to <code>c</code> into an arrow from <code>(b,d)</code> to <code>(c,d)</code>. The idea is that <code>first g</code> uses <code>g</code> to process the first element of a tuple, and lets the second element pass through unchanged. For the function instance of <code>Arrow</code>, of course, <code>first g (x,y) = (g x, y)</code>.<br />
<br />
* The <code>second</code> function is similar to <code>first</code>, but with the elements of the tuples swapped. Indeed, it can be defined in terms of <code>first</code> using an auxiliary function <code>swap</code>, defined by <code>swap (x,y) = (y,x)</code>.<br />
<br />
* The <code>(***)</code> operator is “parallel composition” of arrows: it takes two arrows and makes them into one arrow on tuples, which has the behavior of the first arrow on the first element of a tuple, and the behavior of the second arrow on the second element. The mnemonic is that <code>g *** h</code> is the ''product'' (hence <code>*</code>) of <code>g</code> and <code>h</code>. For the function instance of <code>Arrow</code>, we define <code>(g *** h) (x,y) = (g x, h y)</code>. The default implementation of <code>(***)</code> is in terms of <code>first</code>, <code>second</code>, and sequential arrow composition <code>(>>>)</code>. The reader may also wish to think about how to implement <code>first</code> and <code>second</code> in terms of <code>(***)</code>.<br />
<br />
* The <code>(&&&)</code> operator is “fanout composition” of arrows: it takes two arrows <code>g</code> and <code>h</code> and makes them into a new arrow <code>g &&& h</code> which supplies its input as the input to both <code>g</code> and <code>h</code>, returning their results as a tuple. The mnemonic is that <code>g &&& h</code> performs both <code>g</code> ''and'' <code>h</code> (hence <code>&</code>) on its input. For functions, we define <code>(g &&& h) x = (g x, h x)</code>.<br />
<br />
==Instances==<br />
<br />
The <code>Arrow</code> library itself only provides two <code>Arrow</code> instances, both<br />
of which we have already seen: <code>(->)</code>, the normal function<br />
constructor, and <code>Kleisli m</code>, which makes functions of<br />
type <code>a -> m b</code> into <code>Arrow</code>s for any <code>Monad m</code>. These instances are:<br />
<br />
<haskell><br />
instance Arrow (->) where<br />
arr g = g<br />
first g (x,y) = (g x, y)<br />
<br />
newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b }<br />
<br />
instance Monad m => Arrow (Kleisli m) where<br />
arr f = Kleisli (return . f)<br />
first (Kleisli f) = Kleisli (\ ~(b,d) -> do c <- f b<br />
return (c,d) )<br />
</haskell><br />
<br />
==Laws==<br />
<br />
{{note|See [http://dx.doi.org/10.1016/S0167-6423(99)00023-4 John Hughes: Generalising monads to arrows]; [http://homepages.inf.ed.ac.uk/wadler/papers/arrows/arrows.pdf Sam Lindley, Philip Wadler, Jeremy Yallop: The arrow calculus]; [http://www.soi.city.ac.uk/~ross/papers/fop.html Ross Paterson: Programming with Arrows].}}<br />
<br />
There are quite a few laws that instances of <code>Arrow</code> should<br />
satisfy {{noteref}}:<br />
<br />
<haskell><br />
arr id = id<br />
arr (h . g) = arr g >>> arr h<br />
first (arr g) = arr (g *** id)<br />
first (g >>> h) = first g >>> first h<br />
first g >>> arr (id *** h) = arr (id *** h) >>> first g<br />
first g >>> arr fst = arr fst >>> g<br />
first (first g) >>> arr assoc = arr assoc >>> first g<br />
<br />
assoc ((x,y),z) = (x,(y,z))<br />
</haskell><br />
<br />
Note that this version of the laws is slightly different than the laws given in the<br />
first two above references, since several of the laws have now been<br />
subsumed by the <code>Category</code> laws (in particular, the requirements that<br />
<code>id</code> is the identity arrow and that <code>(>>>)</code> is associative). The laws<br />
shown here follow those in Paterson’s Programming with Arrows, which uses the<br />
<code>Category</code> class.<br />
<br />
{{note|Unless category-theory-induced insomnolence is your cup of tea.}}<br />
<br />
The reader is advised not to lose too much sleep over the <code>Arrow</code><br />
laws {{noteref}}, since it is not essential to understand them in order to<br />
program with arrows. There are also laws that <code>ArrowChoice</code>,<br />
<code>ArrowApply</code>, and <code>ArrowLoop</code> instances should satisfy; the interested<br />
reader should consult [http://www.soi.city.ac.uk/~ross/papers/fop.html Paterson: Programming with Arrows].<br />
<br />
==ArrowChoice==<br />
<br />
Computations built using the <code>Arrow</code> class, like those built using<br />
the <code>Applicative</code> class, are rather inflexible: the structure of the computation<br />
is fixed at the outset, and there is no ability to choose between<br />
alternate execution paths based on intermediate results.<br />
The <code>ArrowChoice</code> class provides exactly such an ability:<br />
<br />
<haskell><br />
class Arrow arr => ArrowChoice arr where<br />
left :: (b `arr` c) -> (Either b d `arr` Either c d)<br />
right :: (b `arr` c) -> (Either d b `arr` Either d c)<br />
(+++) :: (b `arr` c) -> (b' `arr` c') -> (Either b b' `arr` Either c c')<br />
(|||) :: (b `arr` d) -> (c `arr` d) -> (Either b c `arr` d)<br />
</haskell><br />
<br />
A comparison of <code>ArrowChoice</code> to <code>Arrow</code> will reveal a striking<br />
parallel between <code>left</code>, <code>right</code>, <code>(+++)</code>, <code>(|||)</code> and <code>first</code>,<br />
<code>second</code>, <code>(***)</code>, <code>(&&&)</code>, respectively. Indeed, they are dual:<br />
<code>first</code>, <code>second</code>, <code>(***)</code>, and <code>(&&&)</code> all operate on product types<br />
(tuples), and <code>left</code>, <code>right</code>, <code>(+++)</code>, and <code>(|||)</code> are the<br />
corresponding operations on sum types. In general, these operations<br />
create arrows whose inputs are tagged with <code>Left</code> or <code>Right</code>, and can<br />
choose how to act based on these tags.<br />
<br />
* If <code>g</code> is an arrow from <code>b</code> to <code>c</code>, then <code>left g</code> is an arrow from <code>Either b d</code> to <code>Either c d</code>. On inputs tagged with <code>Left</code>, the <code>left g</code> arrow has the behavior of <code>g</code>; on inputs tagged with <code>Right</code>, it behaves as the identity.<br />
<br />
* The <code>right</code> function, of course, is the mirror image of <code>left</code>. The arrow <code>right g</code> has the behavior of <code>g</code> on inputs tagged with <code>Right</code>.<br />
<br />
* The <code>(+++)</code> operator performs “multiplexing”: <code>g +++ h</code> behaves as <code>g</code> on inputs tagged with <code>Left</code>, and as <code>h</code> on inputs tagged with <code>Right</code>. The tags are preserved. The <code>(+++)</code> operator is the ''sum'' (hence <code>+</code>) of two arrows, just as <code>(***)</code> is the product.<br />
<br />
* The <code>(|||)</code> operator is “merge” or “fanin”: the arrow <code>g ||| h</code> behaves as <code>g</code> on inputs tagged with <code>Left</code>, and <code>h</code> on inputs tagged with <code>Right</code>, but the tags are discarded (hence, <code>g</code> and <code>h</code> must have the same output type). The mnemonic is that <code>g ||| h</code> performs either <code>g</code> ''or'' <code>h</code> on its input.<br />
<br />
The <code>ArrowChoice</code> class allows computations to choose among a finite number of execution paths, based on intermediate results. The possible<br />
execution paths must be known in advance, and explicitly assembled with <code>(+++)</code> or <code>(|||)</code>. However, sometimes more flexibility is<br />
needed: we would like to be able to ''compute'' an arrow from intermediate results, and use this computed arrow to continue the computation. This is the power given to us by <code>ArrowApply</code>.<br />
<br />
==ArrowApply==<br />
<br />
The <code>ArrowApply</code> type class is:<br />
<br />
<haskell><br />
class Arrow arr => ArrowApply arr where<br />
app :: (b `arr` c, b) `arr` c<br />
</haskell><br />
<br />
If we have computed an arrow as the output of some previous<br />
computation, then <code>app</code> allows us to apply that arrow to an input,<br />
producing its output as the output of <code>app</code>. As an exercise, the<br />
reader may wish to use <code>app</code> to implement an alternative “curried”<br />
version, <code>app2 :: b `arr` ((b `arr` c) `arr` c)</code>.<br />
<br />
This notion of being able to ''compute'' a new computation<br />
may sound familiar:<br />
this is exactly what the monadic bind operator <code>(>>=)</code> does. It<br />
should not particularly come as a surprise that <code>ArrowApply</code> and<br />
<code>Monad</code> are exactly equivalent in expressive power. In particular,<br />
<code>Kleisli m</code> can be made an instance of <code>ArrowApply</code>, and any instance<br />
of <code>ArrowApply</code> can be made a <code>Monad</code> (via the <code>newtype</code> wrapper<br />
<code>ArrowMonad</code>). As an exercise, the reader may wish to try<br />
implementing these instances:<br />
<br />
<haskell><br />
instance Monad m => ArrowApply (Kleisli m) where<br />
app = -- exercise<br />
<br />
newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)<br />
<br />
instance ArrowApply a => Monad (ArrowMonad a) where<br />
return = -- exercise<br />
(ArrowMonad a) >>= k = -- exercise<br />
</haskell><br />
<br />
==ArrowLoop==<br />
<br />
The <code>ArrowLoop</code> type class is:<br />
<br />
<haskell><br />
class Arrow a => ArrowLoop a where<br />
loop :: a (b, d) (c, d) -> a b c<br />
<br />
trace :: ((b,d) -> (c,d)) -> b -> c<br />
trace f b = let (c,d) = f (b,d) in c<br />
</haskell><br />
<br />
It describes arrows that can use recursion to compute results, and is<br />
used to desugar the <code>rec</code> construct in arrow notation (described<br />
below).<br />
<br />
Taken by itself, the type of the <code>loop</code> method does not seem to tell<br />
us much. Its intention, however, is a generalization of the <code>trace</code><br />
function which is also shown. The <code>d</code> component of the first arrow’s<br />
output is fed back in as its own input. In other words, the arrow<br />
<code>loop g</code> is obtained by recursively “fixing” the second component of<br />
the input to <code>g</code>.<br />
<br />
It can be a bit difficult to grok what the <code>trace</code> function is doing.<br />
How can <code>d</code> appear on the left and right sides of the <code>let</code>? Well,<br />
this is Haskell’s laziness at work. There is not space here for a<br />
full explanation; the interested reader is encouraged to study the<br />
standard <code>fix</code> function, and to read [http://www.soi.city.ac.uk/~ross/papers/fop.html Paterson’s arrow tutorial].<br />
<br />
==Arrow notation==<br />
<br />
Programming directly with the arrow combinators can be painful,<br />
especially when writing complex computations which need to retain<br />
simultaneous reference to a number of intermediate results. With<br />
nothing but the arrow combinators, such intermediate results must be<br />
kept in nested tuples, and it is up to the programmer to remember<br />
which intermediate results are in which components, and to swap,<br />
reassociate, and generally mangle tuples as necessary. This problem<br />
is solved by the special arrow notation supported by GHC, similar to<br />
<code>do</code> notation for monads, that allows names to be assigned to<br />
intermediate results while building up arrow computations. An example<br />
arrow implemented using arrow notation, taken from<br />
Paterson, is:<br />
<br />
<haskell><br />
class ArrowLoop arr => ArrowCircuit arr where<br />
delay :: b -> (b `arr` b)<br />
<br />
counter :: ArrowCircuit arr => Bool `arr` Int<br />
counter = proc reset -> do<br />
rec output <- idA -< if reset then 0 else next<br />
next <- delay 0 -< output + 1<br />
idA -< output<br />
</haskell><br />
<br />
This arrow is intended to<br />
represent a recursively defined counter circuit with a reset line.<br />
<br />
There is not space here for a full explanation of arrow notation; the<br />
interested reader should consult<br />
[http://www.soi.city.ac.uk/~ross/papers/notation.html Paterson’s paper introducing the notation], or his later [http://www.soi.city.ac.uk/~ross/papers/fop.html tutorial which presents a simplified version].<br />
<br />
==Further reading==<br />
<br />
An excellent starting place for the student of arrows is the [http://www.haskell.org/arrows/ arrows web page], which contains an<br />
introduction and many references. Some key papers on arrows include<br />
Hughes’s original paper introducing arrows, [http://dx.doi.org/10.1016/S0167-6423(99)00023-4 Generalising monads to arrows], and [http://www.soi.city.ac.uk/~ross/papers/notation.html Paterson’s paper on arrow notation].<br />
<br />
Both Hughes and Paterson later wrote accessible tutorials intended for a broader<br />
audience: [http://www.soi.city.ac.uk/~ross/papers/fop.html Paterson: Programming with Arrows] and [http://www.cse.chalmers.se/~rjmh/afp-arrows.pdf Hughes: Programming with Arrows].<br />
<br />
Although Hughes’s goal in defining the <code>Arrow</code> class was to<br />
generalize <code>Monad</code>s, and it has been said that <code>Arrow</code> lies “between<br />
<code>Applicative</code> and <code>Monad</code>” in power, they are not directly<br />
comparable. The precise relationship remained in some confusion until<br />
[http://homepages.inf.ed.ac.uk/wadler/papers/arrows-and-idioms/arrows-and-idioms.pdf analyzed by Lindley, Wadler, and Yallop], who<br />
also invented a new calculus of arrows, based on the lambda calculus,<br />
which considerably simplifies the presentation of the arrow laws<br />
(see [http://homepages.inf.ed.ac.uk/wadler/papers/arrows/arrows.pdf The arrow calculus]).<br />
<br />
Some examples of <code>Arrow</code>s include [http://www.haskell.org/yampa/ Yampa], the<br />
[http://www.fh-wedel.de/~si/HXmlToolbox/ Haskell XML Toolkit], and the functional GUI library [[Grapefruit]].<br />
<br />
Some extensions to arrows have been explored; for example, the<br />
[http://www.cs.ru.nl/A.vanWeelden/bi-arrows/ <code>BiArrow</code>s of Alimarine et al.], for two-way instead of one-way<br />
computation.<br />
<br />
The Haskell wiki has [[Research papers/Monads and Arrows|links to many additional research papers relating to <code>Arrow</code>s]].<br />
<br />
=Comonad=<br />
<br />
The final type class we will examine is <code>Comonad</code>. The <code>Comonad</code> class<br />
is the categorical dual of <code>Monad</code>; that is, <code>Comonad</code> is like <code>Monad</code><br />
but with all the function arrows flipped. It is not actually in the<br />
standard Haskell libraries, but it has seen some interesting uses<br />
recently, so we include it here for completeness.<br />
<br />
==Definition==<br />
<br />
The <code>Comonad</code> type class, defined in the <code>Control.Comonad</code> module of<br />
the [http://hackage.haskell.org/package/comonad comonad library], is:<br />
<br />
<haskell><br />
class Functor w => Comonad w where<br />
extract :: f a -> a<br />
<br />
duplicate :: w a -> w (w a)<br />
duplicate = extend id<br />
<br />
extend :: (w a -> b) -> w a -> w b<br />
extend f = fmap f . duplicate<br />
</haskell><br />
<br />
As you can see, <code>extract</code> is the dual of <code>return</code>, <code>duplicate</code> is the dual of <code>join</code>, and <code>extend</code> is the dual of <code>(=<<)</code>. The definition of <code>Comonad</code> is a bit redundant, giving the programmer the choice on whether extend or duplicate are implemented; the other operation then has a default implementation.<br />
<br />
A prototypical example of a <code>Comonad</code> instance is:<br />
<br />
<haskell><br />
-- Infinite lazy streams<br />
data Stream a = Cons a (Stream a)<br />
<br />
-- 'duplicate' is like the list function 'tails'<br />
-- 'extend' computes a new Stream from an old, where the element<br />
-- at position n is computed as a function of everything from<br />
-- position n onwards in the old Stream<br />
instance Comonad Stream where<br />
extract (Cons x _) = x<br />
duplicate s@(Cons x xs) = Cons s (duplicate xs)<br />
extend g s@(Cons x xs) = Cons (g s) (extend g xs)<br />
-- = fmap g (duplicate s)<br />
</haskell><br />
<br />
==Further reading==<br />
<br />
Dan Piponi explains in a blog post what [http://blog.sigfpe.com/2006/12/evaluating-cellular-automata-is.html cellular automata have to do with comonads]. In another blog post, Conal Elliott has examined [http://conal.net/blog/posts/functional-interactive-behavior/ a comonadic formulation of functional reactive programming]. Sterling Clover’s blog post [http://fmapfixreturn.wordpress.com/2008/07/09/comonads-in-everyday-life/ Comonads in everyday life] explains the relationship between comonads and zippers, and how comonads can be used to design a menu system for a web site.<br />
<br />
Uustalu and Vene have a number of papers exploring ideas related to comonads and functional programming:<br />
* [http://dx.doi.org/10.1016/j.entcs.2008.05.029 Comonadic Notions of Computation]<br />
* [http://www.ioc.ee/~tarmo/papers/sfp01-book.pdf The dual of substitution is redecoration] (Also available as [http://www.cs.ut.ee/~varmo/papers/sfp01-book.ps.gz ps.gz].)<br />
* [http://dx.doi.org/10.1016/j.ic.2005.08.005 Recursive coalgebras from comonads]<br />
* [http://www.fing.edu.uy/~pardo/papers/njc01.ps.gz Recursion schemes from comonads]<br />
* [http://cs.ioc.ee/~tarmo/papers/essence.pdf The Essence of Dataflow Programming].<br />
<br />
Gabriel Gonzalez's [http://www.haskellforall.com/2013/02/you-could-have-invented-comonads.html Comonads are objects] points out similarities between comonads and object-oriented programming.<br />
<br />
The [http://hackage.haskell.org/package/comonad-transformers comonad-transformers] package contains comonad transfomers.<br />
<br />
=Acknowledgements=<br />
<br />
A special thanks to all of those who taught me about standard Haskell<br />
type classes and helped me develop good intuition for them,<br />
particularly Jules Bean (quicksilver), Derek Elkins (ddarius), Conal<br />
Elliott (conal), Cale Gibbard (Cale), David House, Dan Piponi<br />
(sigfpe), and Kevin Reid (kpreid).<br />
<br />
I also thank the many people who provided a mountain of helpful<br />
feedback and suggestions on a first draft of the Typeclassopedia: David Amos,<br />
Kevin Ballard, Reid Barton, Doug Beardsley, Joachim Breitner, Andrew<br />
Cave, David Christiansen, Gregory Collins, Mark Jason Dominus, Conal<br />
Elliott, Yitz Gale, George Giorgidze, Steven Grady, Travis Hartwell,<br />
Steve Hicks, Philip Hölzenspies, Edward Kmett, Eric Kow, Serge Le<br />
Huitouze, Felipe Lessa, Stefan Ljungstrand, Eric Macaulay, Rob MacAulay, Simon Meier,<br />
Eric Mertens, Tim Newsham, Russell O’Connor, Conrad Parker, Walt<br />
Rorie-Baety, Colin Ross, Tom Schrijvers, Aditya Siram, C. Smith,<br />
Martijn van Steenbergen, Joe Thornber, Jared Updike, Rob Vollmert,<br />
Andrew Wagner, Louis Wasserman, and Ashley Yakeley, as well as a few<br />
only known to me by their IRC nicks: b_jonas, maltem, tehgeekmeister,<br />
and ziman. I have undoubtedly omitted a few inadvertently, which in<br />
no way diminishes my gratitude.<br />
<br />
Finally, I would like to thank Wouter Swierstra for his fantastic work<br />
editing the Monad.Reader, and my wife Joyia for her patience during<br />
the process of writing the Typeclassopedia.<br />
<br />
=About the author=<br />
<br />
Brent Yorgey ([http://byorgey.wordpress.com/ blog], [http://www.cis.upenn.edu/~byorgey/ homepage]) is (as of November 2011) a fourth-year Ph.D. student in the [http://www.cis.upenn.edu/~plclub/ programming languages group] at the [http://www.upenn.edu University of Pennsylvania]. He enjoys teaching, creating EDSLs, playing Bach fugues, musing upon category theory, and cooking tasty lambda-treats for the denizens of #haskell.<br />
<br />
=Colophon=<br />
<br />
The Typeclassopedia was written by Brent Yorgey and initally published in March 2009. Painstakingly converted to wiki syntax by [[User:Geheimdienst]] in November 2011, after asking Brent’s permission.<br />
<br />
If something like this tex to wiki syntax conversion ever needs to be done again, here are some vim commands that helped:<br />
<br />
* <nowiki>%s/\\section{\([^}]*\)}/=\1=/gc</nowiki><br />
* <nowiki>%s/\\subsection{\([^}]*\)}/==\1==/gc</nowiki><br />
* <nowiki>%s/^ *\\item /\r* /gc</nowiki><br />
* <nowiki>%s/---/—/gc</nowiki><br />
* <nowiki>%s/\$\([^$]*\)\$/<math>\1\\ <\/math>/gc</nowiki> ''Appending “\ ” forces images to be rendered. Otherwise, Mediawiki would go back and forth between one font for short <nowiki><math></nowiki> tags, and another more Tex-like font for longer tags (containing more than a few characters)""<br />
* <nowiki>%s/|\([^|]*\)|/<code>\1<\/code>/gc</nowiki><br />
* <nowiki>%s/\\dots/.../gc</nowiki><br />
* <nowiki>%s/^\\label{.*$//gc</nowiki><br />
* <nowiki>%s/\\emph{\([^}]*\)}/''\1''/gc</nowiki><br />
* <nowiki>%s/\\term{\([^}]*\)}/''\1''/gc</nowiki><br />
<br />
The biggest issue was taking the academic-paper-style citations and turning them into hyperlinks with an appropriate title and an appropriate target. In most cases there was an obvious thing to do (e.g. online PDFs of the cited papers or Citeseer entries). Sometimes, however, it’s less clear and you might want to check the<br />
[[Media:Typeclassopedia.pdf|original Typeclassopedia PDF]]<br />
with the<br />
[http://code.haskell.org/~byorgey/TMR/Issue13/typeclassopedia.bib original bibliography file].<br />
<br />
To get all the citations into the main text, I first tried processing the source with Tex or Lyx. This didn’t work due to missing unfindable packages, syntax errors, and my general ineptitude with Tex.<br />
<br />
I then went for the next best solution, which seemed to be extracting all instances of “\cite{something}” from the source and ''in that order'' pulling the referenced entries from the .bib file. This way you can go through the source file and sorted-references file in parallel, copying over what you need, without searching back and forth in the .bib file. I used:<br />
<br />
* <nowiki>egrep -o "\cite\{[^\}]*\}" ~/typeclassopedia.lhs | cut -c 6- | tr "," "\n" | tr -d "}" > /tmp/citations</nowiki><br />
* <nowiki>for i in $(cat /tmp/citations); do grep -A99 "$i" ~/typeclassopedia.bib|egrep -B99 '^\}$' -m1 ; done > ~/typeclasso-refs-sorted</nowiki><br />
<br />
[[Category:Applicative Functor]]<br />
[[Category:Arrow]]<br />
[[Category:Functor]]<br />
[[Category:Monad]]<br />
[[Category:Standard classes]]<br />
[[Category:Standard libraries]]<br />
[[Category:Standard packages]]<br />
[[Category:Standard types]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56316Functor-Applicative-Monad Proposal2013-06-21T16:49:45Z<p>Blackout: /* Prepare GHC */ Marked as "current stage"</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
Future versions of GHC will issue warnings if code doesn't comply to these rules; there will be a long enough transitional phase so Hackage can adapt to the AMP in advance before the above mentioned changes are actually enforced.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== '''Current stage''': Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56313Functor-Applicative-Monad Proposal2013-06-21T13:37:37Z<p>Blackout: </p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the very end). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
Future versions of GHC will issue warnings if code doesn't comply to these rules; there will be a long enough transitional phase so Hackage can adapt to the AMP in advance before the above mentioned changes are actually enforced.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56312Functor-Applicative-Monad Proposal2013-06-21T13:36:46Z<p>Blackout: Removed invalid ticket link</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: ticket [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
Future versions of GHC will issue warnings if code doesn't comply to these rules; there will be a long enough transitional phase so Hackage can adapt to the AMP in advance before the above mentioned changes are actually enforced.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56310Functor-Applicative-Monad Proposal2013-06-21T11:33:46Z<p>Blackout: Added links to the tickets</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
Some relevant links:<br />
* [https://github.com/quchen/articles/blob/master/applicative_monad.md Initial text of the Haskell 2014 AMP]<br />
* [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 AMP mailing list discussion]<br />
* Phase one: tickets [http://hackage.haskell.org/trac/ghc/ticket/8003 #8003] and [http://hackage.haskell.org/trac/ghc/ticket/8004 #8004]<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
Future versions of GHC will issue warnings if code doesn't comply to these rules; there will be a long enough transitional phase so Hackage can adapt to the AMP in advance before the above mentioned changes are actually enforced.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56309Functor-Applicative-Monad Proposal2013-06-21T08:26:43Z<p>Blackout: Small corrections, markup etc.</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
[https://github.com/quchen/articles/blob/master/applicative_monad.md The initial text of the Haskell 2014 AMP can be found here], and [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 here's the mailing list discussion of the proposal].<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
Future versions of GHC will issue warnings if code doesn't comply to these rules; there will be a long enough transitional phase so Hackage can adapt to the AMP in advance before the above mentioned changes are actually enforced.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is potentially the most laborious consequence. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
* Monad without Applicative<br />
* MonadPlus without Alternative<br />
* One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* Early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; changes similar to this proposal, but closed as "not GHC, but Haskell". See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905 here] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56287Functor-Applicative-Monad Proposal2013-06-18T10:11:40Z<p>Blackout: Fixed list numbering</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
[https://github.com/quchen/articles/blob/master/applicative_monad.md The initial text of the Haskell 2014 AMP can be found here], and [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 here's the mailing list discussion of the proposal].<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
Future versions of GHC will issue warnings if code doesn't comply to these rules; there will be a long enough transitional phase so Hackage can adapt to the AMP in advance before the above mentioned changes are actually enforced.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
# Libraries using their own <hask>(<*>)</hask>. This one is much tougher, as renaming the operator may require a lot of effort. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
- Monad without Applicative<br />
- MonadPlus without Alternative<br />
- One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* From early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; Makes Applicative into a superclass of Monad, but does not deprecate any existing names<br />
** See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56286Functor-Applicative-Monad Proposal2013-06-18T09:01:01Z<p>Blackout: /* How do programmers need to change their code? */ Comment about transitional phase</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
[https://github.com/quchen/articles/blob/master/applicative_monad.md The initial text of the Haskell 2014 AMP can be found here], and [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 here's the mailing list discussion of the proposal].<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
Future versions of GHC will issue warnings if code doesn't comply to these rules; there will be a long enough transitional phase so Hackage can adapt to the AMP in advance before the above mentioned changes are actually enforced.<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
<br />
# Libraries using their own <hask>(<*>)</hask>. This one is much tougher, as renaming the operator may require a lot of effort. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
- Monad without Applicative<br />
- MonadPlus without Alternative<br />
- One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* From early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; Makes Applicative into a superclass of Monad, but does not deprecate any existing names<br />
** See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56285Functor-Applicative-Monad Proposal2013-06-18T08:56:03Z<p>Blackout: Finished porting the AMP from Github to the Wiki. This page should be self-contained now.</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
[https://github.com/quchen/articles/blob/master/applicative_monad.md The initial text of the Haskell 2014 AMP can be found here], and [http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 here's the mailing list discussion of the proposal].<br />
<br />
<br />
<br />
== Proposal contents ==<br />
<br />
The list of changes is as follows:<br />
<br />
# Applicative becomes a superclass of Monad, and is added to the Prelude.<br />
# Alternative becomes a superclass of MonadPlus (in addition to Monad, of course).<br />
# <hask>join</hask> is promoted into the Monad typeclass.<br />
<br />
The general rationale behind these changes:<br />
<br />
# ''Break as little code as possible.'' For example, do not move <hask>return</hask> to Applicative and remove <hask>pure</hask>. Instead, leave <hask>return</hask> in Monad, and give it <hask>pure</hask> as default implementation.<br />
<br />
# ''Change only things that are closely related to the proposal.'' For example, using <hask>join</hask> in a monad definition requires it to be a functor, so it goes hand in hand with the AMP. On the other hand, removing <hask>fail</hask> has nothing to do with what we're trying to accomplish.<br />
<br />
<br />
<br />
== How do programmers need to change their code? ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
<br />
<br />
== Discussion and consequences ==<br />
<br />
<br />
=== It's the right thing to do™ ===<br />
<br />
Math. You've all heard this one, it's good and compelling so I don't need to spell it out.<br />
<br />
<br />
=== Redundant functions ===<br />
<br />
* <hask>pure</hask> and <hask>return</hask> do the same thing.<br />
* <hask>>></hask> and <hask>*></hask> are identical.<br />
* <hask>liftM</hask> and <hask>liftA</hask> are <hask>fmap</hask>. The <hask>liftM*</hask> are <hask>liftA*</hask>, <hask><*></hask> is <hask>ap</hask>.<br />
* Prelude's <hask>sequence</hask> requres <hask>Monad</hask> right now, while <hask>Applicative</hask> is sufficient to implement it. The more general version of this issue is captured by <hask>Data.Traversable</hask>, whose main typeclass implements the *same* functionality twice, namely <hask>traverse</hask> and <hask>mapM</hask>, and <hask>sequenceA</hask> and <hask>sequence</hask>.<br />
* The <hask>WrappedMonad</hask> type from <hask>Control.Applicative</hask> provides a semi-automatic way to using Functor/Applicative/Alternative functions for Monad/MonadPlus instances as a makeshift patch.<br />
<br />
That very much violates the "don't repeat yourself" principle, and even more so it ''forces'' the programmer to repeat himself to achieve maximal generality. It may be too late to take all redundancies out, but at least we can prevent new ones from being created.<br />
<br />
(Note that it is not proposed to remove any functions for compatibility reasons. Maybe some of them can be phased out in the long run, but that's beyond scope here.)<br />
<br />
<br />
=== Using Functor/Applicative functions in monadic code ===<br />
<br />
Whenever there's Monad code, you can use Functor/Applicative functions, without introducing an additional constraint. Keep in mind that "Functor/Applicative functions" does not only include what their typeclasses define but many more, for example <hask>void</hask>, <hask>(<$>)</hask>, <hask>(<**>)</hask>.<br />
<br />
Even if you think you have monadic code, strictly using the least restrictive functions may result in something that requires only Applicative. This is similar to writing a function that needs <hask>Int</hask>, but it turns out any <hask>Integral</hask> will do - more polymorphism for free.<br />
<br />
<br />
=== Compatibility issues ===<br />
<br />
These are the kinds of issues to be expected:<br />
<br />
# Monads lacking Functor or Applicative instances. This is easily fixable by either setting <hask>fmap = liftM</hask>, <hask>pure = return</hask> and <hask>(<*>) = ap</hask>, although more efficient implementations may exist, or by moving an already existing definition from <hask>Control.Applicative</hask> to the appropriate module.<br />
<br />
# This one is specific to building GHC: importing <hask>Control.Monad/Applicative</hask> introduces a circular module dependency. In this case, one can rely on handwritten implementations of the desired function, e.g. <hask>ap f x = f >>= ...</hask>.<br />
<br />
# Libraries using their own <hask>(<*>)</hask>. This one is much tougher, as renaming the operator may require a lot of effort. For building GHC though, this only concerns Hoopl, and a handful of renames.<br />
<br />
<br />
<br />
=== Beginner friendliness ===<br />
<br />
How often did you say ...<br />
<br />
* "A Monad is always an Applicative but due to historical reasons it's not but you can easily verify it by setting <hask>pure = return</hask> and <hask>(<*>) = ap</hask>"<br />
* "<hask>liftM</hask> is <hask>fmap</hask> but not really." - "So when should I use <hask>fmap</hask> and when <hask>liftM</hask>?" - *sigh*<br />
<br />
With the new hierarchy, the answer would *always* be "use the least restrictive one".<br />
<br />
<br />
<br />
== Applying the AMP to GHC and then Haskell in practice ==<br />
<br />
Proposed is a gradual introduction of the AMP in three phases:<br />
<br />
<br />
=== Prepare GHC ===<br />
<br />
Using a GHC fork with the full patch applied, find and fix all compilation errors introduced by the change by adding Functor/Applicative instances for all Monads.<br />
<br />
According to SPJ, adding an ad-hoc warning of sorts "Monad without Applicative detected" is not a problem, which will be crucial for the next phase. More specifically, issue a warning if:<br />
<br />
- Monad without Applicative<br />
- MonadPlus without Alternative<br />
- One of <hask><*></hask>, <hask>pure</hask>, <hask>join</hask> is defined in a different context to avoid naming conflicts, as these functions will go into the Prelude<br />
<br />
<br />
=== Prepare Hackage ===<br />
<br />
The warning just mentioned will hint to all authors that they should fix (or help others fix) the non-complying packages. This will ideally lead to libraries eventually adding Applicative instances, and changing their APIs if they redefine operators like <hask><*></hask>.<br />
<br />
After enough time has passed by so libraries adapted to the circumstances, move on to the next phase.<br />
<br />
<br />
=== Apply the proposal ===<br />
<br />
Once Hackage is prepared, applying the changes to the Base package is painless. However, this is not primarily a GHC, but a Haskell change. The previous steps were basically preparing the landscape, and when we've (hopefully) found out that it is a good idea to go through with it, it can be proposed to go into the Report. If we make it this far, the AMP should pass quite easily.<br />
<br />
<br />
<br />
== Previous proposals ==<br />
<br />
* From early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; Makes Applicative into a superclass of Monad, but does not deprecate any existing names<br />
** See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56280Functor-Applicative-Monad Proposal2013-06-17T13:03:12Z<p>Blackout: typos</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal (AMP)'''.<br />
<br />
== Pragmatic and short version ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask>.<br />
<br />
== Detailed description ==<br />
<br />
[https://github.com/quchen/articles/blob/master/applicative_monad.md Main text of the Haskell 2014 AMP]<br />
<br />
[http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 Mailing list discussion of the proposal]<br />
<br />
== Previous proposals ==<br />
<br />
* From early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; Makes Applicative into a superclass of Monad, but does not deprecate any existing names<br />
** See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Functor-Applicative-Monad_Proposal&diff=56279Functor-Applicative-Monad Proposal2013-06-17T12:58:41Z<p>Blackout: Updated for AMP 2014</p>
<hr />
<div>Haskell calls a couple of historical accidents its own. While some of them, such as the "number classes" hierarchy, can be justified by pragmatism or lack of a strictly better suggestion, there is one thing that stands out as, well, not that: Applicative not being a superclass of Monad.<br />
<br />
The topic has been discussed multiple times in the past (cf. link section at the bottom). '''This article was updated to describe the current, and very likely to succeed, Haskell 2014 Applicative => Monad proposal'''.<br />
<br />
== Pragmatic and short version ==<br />
<br />
The following is a list of things you may have to change in your code so the AMP doesn't break it.<br />
<br />
* Add Applicative/Functor instances for all your Monads. If you don't care about efficiency, you can simply derive these instances from the Monad by adding the following code:<br />
<haskell><br />
-- Monad m<br />
<br />
import Control.Monad (liftM, ap)<br />
import Control.Applicative (Applicative(..))<br />
<br />
instance Functor m where<br />
fmap = liftM<br />
<br />
instance Applicative m where<br />
pure = return<br />
(<*>) = ap<br />
</haskell><br />
<br />
* Add an Alternative instance for all instances of MonadZero. This can again be done easily using<br />
<haskell><br />
-- MonadZero m<br />
<br />
import Control.Monad (mzero, mplus)<br />
import Control.Applicative (Alternative(..))<br />
<br />
instance Alternative m where<br />
(<|>) = mplus<br />
empty = mzero<br />
</haskell><br />
<br />
* Change your API to not define functions named <hask><*></hask>, <hask>join</hask> or <hask>pure</hask><br />
<br />
== Detailed description ==<br />
<br />
[https://github.com/quchen/articles/blob/master/applicative_monad.md Main text of the Haskell 2014 AMP]<br />
<br />
[http://article.gmane.org/gmane.comp.lang.haskell.libraries/19482 Mailing list discussion of the proposal]<br />
<br />
== Previous proposals ==<br />
<br />
* From early 2011: [http://hackage.haskell.org/trac/ghc/ticket/4834 GHC ticket] &ndash; Makes Applicative into a superclass of Monad, but does not deprecate any existing names<br />
** See [http://thread.gmane.org/gmane.comp.lang.haskell.libraries/14883/focus=14905] for the associated discussion.<br />
* [[The Other Prelude]]<br />
<br />
<br />
[[Category:Proposals]]<br />
[[Category:Functor]]<br />
[[Category:Applicative Functor]]<br />
[[Category:Monad]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Lazy_pattern_match&diff=55525Lazy pattern match2013-03-06T14:51:17Z<p>Blackout: Corrected line/paragraph breaks</p>
<hr />
<div>What does "lazy pattern match" mean<br />
and what is the meaning of the tilde in pattern matches?<br />
<br />
== Syntax ==<br />
<br />
These are all lazy pattern matches:<br />
<haskell><br />
let (a,b) = p<br />
</haskell><br />
<haskell><br />
f ~(a,b) = ...<br />
</haskell><br />
<haskell><br />
case p of ~(a,b) -> ...<br />
</haskell><br />
<haskell><br />
(\ ~(a,b) -> ... )<br />
</haskell><br />
The <hask>let</hask> matches the top-most constructor lazily.<br />
This seems to be quite arbitrary but this is how it is defined.<br />
That is, if you want to match constructors lazily in two levels<br />
then you have to write:<br />
<haskell><br />
let (a, ~(b,c)) = p<br />
</haskell><br />
<haskell><br />
f ~(a, ~(b,c)) = ...<br />
</haskell><br />
<haskell><br />
case p of ~(a, ~(b,c)) -> ...<br />
</haskell><br />
<haskell><br />
(\ ~(a, ~(b,c)) -> ... )<br />
</haskell><br />
<br />
<br />
== Meaning ==<br />
<br />
What is the meaning of a lazy pattern match<br />
and why is it required sometimes?<br />
<br />
The lazy pattern match on a pair as in<br />
<haskell><br />
f ~(a,b) = g a b<br />
</haskell><br />
can be translated to<br />
<haskell><br />
f p = g (fst p) (snd p)<br />
</haskell><br />
Generally, a lazy pattern match is translated to calling<br />
corresponding record field accessors. The key difference between strict pattern match<br />
<haskell><br />
f (a,b) = g a b<br />
</haskell><br />
and lazy pattern match<br />
<haskell><br />
f ~(a,b) = g a b<br />
</haskell><br />
is that the strict pattern match requires to check for the pair constructor before <hask>g</hask> can be evaluated. In contrast to that, the lazy pattern match allows to defer the pair constructor match to the evaluation of <hask>g a b</hask>. If the function <hask>g</hask> can generate something without looking at its arguments then <hask>f</hask> can generate something as well before matching the pair constructor.<br />
<br />
This difference can be essential. For example compare the implementation of <hask>splitAt</hask> with lazy and strict pattern match. Here is the (correct) lazy implementation:<br />
<haskell><br />
import Prelude hiding (splitAt)<br />
<br />
splitAt :: Int -> [a] -> ([a], [a])<br />
splitAt n xs =<br />
if n<=0<br />
then ([], xs)<br />
else<br />
case xs of<br />
[] -> ([], [])<br />
y:ys -><br />
case splitAt (n-1) ys of<br />
~(prefix, suffix) -> (y : prefix, suffix)<br />
</haskell><br />
Now try<br />
<haskell><br />
Test> splitAt 1000000 $ repeat 'a'<br />
</haskell><br />
With lazy pattern match in the last line of the <hask>splitAt</hask> implementation you see an answer immediately whereas with a strict pattern match the Haskell interpreter requires some time and memory before showing something. The reason is that the strict pattern match forces the interpreter to perform all recursive calls to <hask>splitAt</hask> in order to check whether they actually generate a pair constructor. This might look silly since the pair type has only one constructor but mind you that a pair value can also be <hask>undefined</hask>. Furthermore this behavior is consistent with data types with more than one constructor.<br />
<br />
If you are uncertain how that <hask>splitAt</hask> magic works then it might help to translate the tilde into pair element accessors <hask>fst</hask> and <hask>snd</hask>.<br />
<br />
<br />
== Implications ==<br />
<br />
The lazy pattern match has some consequences. First of all a lazy pattern matches immediately always. Remember,<br />
<haskell><br />
f ~(x:xs) = x:xs<br />
</haskell><br />
is translated to<br />
<haskell><br />
f ys = head ys : tail ys<br />
</haskell><br />
and the match on <hask>ys</hask> always succeeds. That is with the lazy pattern match the order of matches counts, e.g.<br />
<haskell><br />
f :: [a] -> [a]<br />
f [] = []<br />
f ~(x:xs) = x:xs<br />
</haskell><br />
is fine but stupid, because the first match already requires the decision whether the list is empty or not. But the reversed order<br />
<haskell><br />
f :: [a] -> [a]<br />
f ~(x:xs) = x:xs<br />
f [] = []<br />
</haskell><br />
yields a compiler warning for an unnecessary match on <hask>[]</hask> because the first case already matches always.<br />
<br />
You may have also noted the danger caused by a lazy pattern match. Since the lazy pattern match on a non-empty list is translated to <hask>head</hask> and <hask>tail</hask> it is as dangerous as <hask>head</hask> and <hask>tail</hask>, because these functions are not [[total function|total]]. That is, generally it is good style to avoid lazy pattern matches on types with more than one constructor.<br />
<br />
<br />
[[Category:Glossary]]<br />
[[Category:Idioms]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Lambdabot&diff=55448Lambdabot2013-02-16T22:37:52Z<p>Blackout: Added two more entries to the 'useful commands' list</p>
<hr />
<div>[[Image:lambdabot.png|Lambdabot]]<br />
<br />
Lambdabot is an IRC bot written over several years by those on the #haskell<br />
[[IRC channel]]. It also operates in an offline mode as a Haskell development tool, and embedded as an extension to ghci.<br />
<br />
Some useful commands for <code>#haskell</code>'s Lambdabot are<br />
* [[Haskell]] evaluation: <code>> take 5 [1..] ==> [1,2,3,4,5]</code><br />
* Defining persistent bindings for evaluation: <code>@let x = 4</code> sets <code>x</code> to 4. <code>@undef</code> deletes old definitions.<br />
* Asking for a functions type, like GHCi's <code>:t</code>, using either that or <code>@ty</code>, <code>@type</code><br />
* Using Hoogle: <code>@hoogle [a] -> [a]</code><br />
* [[Pointfree]] refactoring: <code>@pl \xs n -> take n xs ==> flip take</code><br />
* Djinn takes a type signature and constructs a function satisfying it: <code>@djinn (a, b) -> c -> (b, c) ==> f (_, a) b = (a, b)</code><br />
* What type does a monad transformer wrap? <code>@unmtl StateT s IO a ==> s -> IO (a, s)</code><br />
* Convert do notation to bind notation: <code>@undo do { x <- getLine; print x } ==> getLine >>= \ x -> print x</code><br />
* Lambdabot has a huge list of imports. If it can't find the name of a function you specified, try the fully qualified name. There is no way of importing modules manually.<br />
<br />
<br />
=== Features ===<br />
<br />
Lambdabot is written in Haskell ([http://darcs.net Darcs] repository [http://code.haskell.org/lambdabot here]), and supports plugins for adding new<br />
commands. It has many plugins, including:<br />
<br />
* [[Haskell]] evaluation (with persistent declarations)<br />
* [[Hoogle]] interface<br />
* [[Pointfree]] refactoring<br />
* A theorem prover, [[Libraries and tools/Theorem_provers|Djinn]].<br />
* Two [http://homepages.inf.ed.ac.uk/wadler/papers/free/free.ps free theorems] generators<br />
* A testing rig using [[QuickCheck]]<br />
* Another test rig using [[SmallCheck]]<br />
* A lambda calculus interpreter<br />
* A Unlambda interpreter<br />
* A Brainfuck interpreter<br />
* Haskell type and kind checking<br />
* Dynamic plugin composition<br />
* Haskell [http://darcs.haskell.org/packages/ library source] lookup<br />
* Language translation<br />
* [http://code.haskell.org/lambdabot/State/quote Quotes]<br />
* Todo lists<br />
* Irc functions<br />
* [[Darcs]] patch tracking<br />
* A vixen/eliza personality<br />
* Random dice<br />
* [http://code.haskell.org/lambdabot/State/where Project database]<br />
* Dictionary lookups<br />
* Karma tracking<br />
* User poll and election support<br />
* Search Google, Wikipedia and more<br />
* Spell checking<br />
* [http://tinyurl.com Tiny URL] generation<br />
* Much more...<br />
<br />
=== Installing ===<br />
If you are using GHC 6.10.x, just <tt>[[cabal install]] lambdabot</tt>. 6.12.x support is forthcoming; see [[/Building]] for details on that. (It is possible that Hackage lambdabot will just work with 6.12.x.)<br />
<br />
=== Interacting with the Bot ===<br />
<br />
Almost all commands start with an <TT>@</TT> or <tt>?</tt>. A command is invoked by<br />
writing <TT>@<command-name> [args]</TT>. For example <TT>@type map</TT><br />
will make lambdabot respond with <hask>(a -> b) -> [a] -> [b]</hask>. <br />
<br />
The special command <tt>@run</tt> has a synonym <tt>></tt>, meaning that<br />
you can evaluate Haskell expressions dynamically with:<br />
<br />
<haskell><br />
> map (+1) [1..10]<br />
[2,3,4,5,6,7,8,9,10,11]<br />
</haskell><br />
<br />
Good commands to know are <tt>@help</tt> and <tt>@list</tt>.<br />
The full command list is [http://code.haskell.org/lambdabot/COMMANDS here].<br />
<br />
=== Contributing ===<br />
<br />
Lambdabot is an open-source application. The <tt>@version</tt> command<br />
provides details on where to find the source. Anyone may write a plugin.<br />
To submit a plugin, use ''darcs send'' to submit it to the bot's current<br />
maintainer.<br />
<br />
For more info, see the sources or ask the bot itself. If you have any<br />
questions about Lambdabot, ask on the #haskell irc channel.<br />
<br />
=== How can it securely execute Haskell, on the Web? ===<br />
<br />
See [[Safely running untrusted Haskell code]]<br />
<br />
=== @version ===<br />
<br />
A list of various lambdabots over time.<br />
#haskell lambdabot 4p534, GHC 6.6 (Linux i686 2.66GHz)<br />
#haskell lambdabot 4p255, GHC 6.5 (OpenBSD i386)<br />
#haskell lambdabot 4p172, GHC 6.5 (OpenBSD i386)<br />
#haskell lambdabot 3p396, GHC 6.4.1 (Linux i686 3.20GHz)<br />
#haskell lambdabot 2p200, GHC 6.4 (OpenBSD i386)<br />
#haskell lambdabot ghc-6.2.2 (Mar 30 20:48:23 EST 2005) (OpenBSD i386)<br />
#maya lambdabot 3.1p48, GHC 6.5 (OpenBSD i386)<br />
#dutchhack (irc.cyberarmy.net) lambdabot 3p399, GHC 6.4.1 (Linux sparc64)<br />
<br />
=== Lambdabot 2 ===<br />
<br />
Sketch of design for a new lambdabot, smaller, more maintainable, more<br />
reliable. Features:<br />
<br />
Core:<br />
* highly concurrent, as for lambdabot 1<br />
* multiple servers<br />
* configuration scripts<br />
* reliable, robust state checkpointing<br />
* command line, irc, and ide interface<br />
* simple plugin api<br />
* single cabalised bundle<br />
<br />
New plugins:<br />
* beta reduction/term rewriting<br />
* hackage search<br />
* rss subscription<br />
<br />
Improved plugins:<br />
* Hoogle<br />
<br />
[[Category:Tools]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Lambdabot&diff=55447Lambdabot2013-02-16T22:33:52Z<p>Blackout: Added "useful commands" section</p>
<hr />
<div>[[Image:lambdabot.png|Lambdabot]]<br />
<br />
Lambdabot is an IRC bot written over several years by those on the #haskell<br />
[[IRC channel]]. It also operates in an offline mode as a Haskell development tool, and embedded as an extension to ghci.<br />
<br />
Some useful commands for <code>#haskell</code>'s Lambdabot are<br />
* [[Haskell]] evaluation: <code>> take 5 [1..] ==> [1,2,3,4,5]</code><br />
* Defining persistent bindings for evaluation: <code>@let x = 4</code> sets <code>x</code> to 4. <code>@undef</code> deletes old definitions.<br />
* Using Hoogle: <code>@hoogle [a] -> [a]</code><br />
* [[Pointfree]] refactoring: <code>@pl \xs n -> take n xs ==> flip take</code><br />
* Djinn takes a type signature and constructs a function satisfying it: <code>@djinn (a, b) -> c -> (b, c) ==> f (_, a) b = (a, b)</code><br />
* What type does a monad transformer wrap? <code>@unmtl StateT s IO a ==> s -> IO (a, s)</code><br />
* Convert do notation to bind notation: <code>@undo do { x <- getLine; print x } ==> getLine >>= \ x -> print x</code><br />
<br />
<br />
=== Features ===<br />
<br />
Lambdabot is written in Haskell ([http://darcs.net Darcs] repository [http://code.haskell.org/lambdabot here]), and supports plugins for adding new<br />
commands. It has many plugins, including:<br />
<br />
* [[Haskell]] evaluation (with persistent declarations)<br />
* [[Hoogle]] interface<br />
* [[Pointfree]] refactoring<br />
* A theorem prover, [[Libraries and tools/Theorem_provers|Djinn]].<br />
* Two [http://homepages.inf.ed.ac.uk/wadler/papers/free/free.ps free theorems] generators<br />
* A testing rig using [[QuickCheck]]<br />
* Another test rig using [[SmallCheck]]<br />
* A lambda calculus interpreter<br />
* A Unlambda interpreter<br />
* A Brainfuck interpreter<br />
* Haskell type and kind checking<br />
* Dynamic plugin composition<br />
* Haskell [http://darcs.haskell.org/packages/ library source] lookup<br />
* Language translation<br />
* [http://code.haskell.org/lambdabot/State/quote Quotes]<br />
* Todo lists<br />
* Irc functions<br />
* [[Darcs]] patch tracking<br />
* A vixen/eliza personality<br />
* Random dice<br />
* [http://code.haskell.org/lambdabot/State/where Project database]<br />
* Dictionary lookups<br />
* Karma tracking<br />
* User poll and election support<br />
* Search Google, Wikipedia and more<br />
* Spell checking<br />
* [http://tinyurl.com Tiny URL] generation<br />
* Much more...<br />
<br />
=== Installing ===<br />
If you are using GHC 6.10.x, just <tt>[[cabal install]] lambdabot</tt>. 6.12.x support is forthcoming; see [[/Building]] for details on that. (It is possible that Hackage lambdabot will just work with 6.12.x.)<br />
<br />
=== Interacting with the Bot ===<br />
<br />
Almost all commands start with an <TT>@</TT> or <tt>?</tt>. A command is invoked by<br />
writing <TT>@<command-name> [args]</TT>. For example <TT>@type map</TT><br />
will make lambdabot respond with <hask>(a -> b) -> [a] -> [b]</hask>. <br />
<br />
The special command <tt>@run</tt> has a synonym <tt>></tt>, meaning that<br />
you can evaluate Haskell expressions dynamically with:<br />
<br />
<haskell><br />
> map (+1) [1..10]<br />
[2,3,4,5,6,7,8,9,10,11]<br />
</haskell><br />
<br />
Good commands to know are <tt>@help</tt> and <tt>@list</tt>.<br />
The full command list is [http://code.haskell.org/lambdabot/COMMANDS here].<br />
<br />
=== Contributing ===<br />
<br />
Lambdabot is an open-source application. The <tt>@version</tt> command<br />
provides details on where to find the source. Anyone may write a plugin.<br />
To submit a plugin, use ''darcs send'' to submit it to the bot's current<br />
maintainer.<br />
<br />
For more info, see the sources or ask the bot itself. If you have any<br />
questions about Lambdabot, ask on the #haskell irc channel.<br />
<br />
=== How can it securely execute Haskell, on the Web? ===<br />
<br />
See [[Safely running untrusted Haskell code]]<br />
<br />
=== @version ===<br />
<br />
A list of various lambdabots over time.<br />
#haskell lambdabot 4p534, GHC 6.6 (Linux i686 2.66GHz)<br />
#haskell lambdabot 4p255, GHC 6.5 (OpenBSD i386)<br />
#haskell lambdabot 4p172, GHC 6.5 (OpenBSD i386)<br />
#haskell lambdabot 3p396, GHC 6.4.1 (Linux i686 3.20GHz)<br />
#haskell lambdabot 2p200, GHC 6.4 (OpenBSD i386)<br />
#haskell lambdabot ghc-6.2.2 (Mar 30 20:48:23 EST 2005) (OpenBSD i386)<br />
#maya lambdabot 3.1p48, GHC 6.5 (OpenBSD i386)<br />
#dutchhack (irc.cyberarmy.net) lambdabot 3p399, GHC 6.4.1 (Linux sparc64)<br />
<br />
=== Lambdabot 2 ===<br />
<br />
Sketch of design for a new lambdabot, smaller, more maintainable, more<br />
reliable. Features:<br />
<br />
Core:<br />
* highly concurrent, as for lambdabot 1<br />
* multiple servers<br />
* configuration scripts<br />
* reliable, robust state checkpointing<br />
* command line, irc, and ide interface<br />
* simple plugin api<br />
* single cabalised bundle<br />
<br />
New plugins:<br />
* beta reduction/term rewriting<br />
* hackage search<br />
* rss subscription<br />
<br />
Improved plugins:<br />
* Hoogle<br />
<br />
[[Category:Tools]]</div>Blackouthttps://wiki.haskell.org/index.php?title=GADTs_for_dummies&diff=55404GADTs for dummies2013-02-09T19:08:43Z<p>Blackout: No wiki should house a section called "Random rubbish from previous versions of article". Deleted.</p>
<hr />
<div>[[Category:Tutorials]]<br />
[[Category:Language extensions]]<br />
<br />
For a long time, I didn't understand what GADTs were or how they could be used. It was sort of a conspiracy of silence - people who understood GADTs thought<br />
that everything was obvious and didn't need further explanation, but I still<br />
couldn't understand them.<br />
<br />
Now that I have an idea how it works, I think that it was really obvious. :) So, I want to share my understanding GADTs. Maybe the way I realized how GADTs work could help<br />
someone else. See also [[Generalised algebraic datatype]]<br />
<br />
== Type functions ==<br />
<br />
A "data" declaration is a way to declare both type constructor and data<br />
constructors. For example,<br />
<br />
<haskell><br />
data Either a b = Left a | Right b<br />
</haskell><br />
<br />
declares type constructor "Either" and two data constructors "Left"<br />
and "Right". Ordinary Haskell functions works with data constructors:<br />
<br />
<haskell><br />
isLeft (Left a) = True<br />
isLeft (Right b) = False<br />
</haskell><br />
<br />
but there is also an analogous way to work with type constructors!<br />
<br />
<haskell><br />
type X a = Either a a<br />
</haskell><br />
<br />
declares a TYPE FUNCTION named "X". Its parameter "a" must be some type<br />
and it returns some type as its result. We can't use "X" on data values,<br />
but we can use it on type values. Type constructors declared with<br />
"data" statements and type functions declared with "type" statements<br />
can be used together to build arbitrarily complex types. In such<br />
"computations" type constructors serves as basic "values" and type<br />
functions as a way to process them.<br />
<br />
Indeed, type functions in Haskell are very limited compared to<br />
ordinary functions - they don't support pattern matching,<br />
nor multiple statements, nor recursion.<br />
<br />
== Hypothetical Haskell extension - Full-featured type functions ==<br />
<br />
Let's build a hypothetical Haskell extension that mimics, for type<br />
functions, the well-known ways to define ordinary functions, including<br />
pattern matching:<br />
<br />
<haskell><br />
type F [a] = Set a<br />
</haskell><br />
<br />
multiple statements (this is meaningful only in the presence of pattern matching):<br />
<br />
<haskell><br />
type F Bool = Char<br />
F String = Int<br />
</haskell><br />
<br />
and recursion (which again needs pattern matching and multiple statements):<br />
<br />
<haskell><br />
type F [a] = F a<br />
F (Map a b) = F b<br />
F (Set a) = F a<br />
F a = a<br />
</haskell><br />
<br />
As you may already have guessed, this last definition calculates a simple base type of arbitrarily-nested collections, e.g.: <br />
<br />
<haskell><br />
F [[[Set Int]]] = <br />
F [[Set Int]] =<br />
F [Set Int] = <br />
F (Set Int) =<br />
F Int =<br />
Int<br />
</haskell><br />
<br />
Let's not forget about statement guards:<br />
<br />
<haskell><br />
type F a | IsSimple a == TrueType = a<br />
</haskell><br />
<br />
Here we define type function F only for simple datatypes by using a<br />
guard type function "IsSimple":<br />
<br />
<haskell><br />
type IsSimple Bool = TrueType<br />
IsSimple Int = TrueType<br />
....<br />
IsSimple Double = TrueType<br />
IsSimple a = FalseType<br />
<br />
data TrueType = T<br />
data FalseType = F<br />
</haskell><br />
<br />
These definitions seem a bit odd, and while we are in imaginary land,<br />
let's consider a way to write this shorter:<br />
<br />
<haskell><br />
type F a | IsSimple a = a<br />
<br />
type IsSimple Bool<br />
IsSimple Int<br />
....<br />
IsSimple Double<br />
</haskell><br />
<br />
Here, we defined a list of simple types. The implied result of all<br />
written statements for "IsSimple" is True, and False for<br />
everything else. Essentially, "IsSimple" is a TYPE PREDICATE!<br />
<br />
I really love it! :) How about constructing a predicate that traverses a<br />
complex type trying to decide whether it contains "Int" anywhere?<br />
<br />
<haskell><br />
type HasInt Int<br />
HasInt [a] = HasInt a<br />
HasInt (Set a) = HasInt a<br />
HasInt (Map a b) | HasInt a<br />
HasInt (Map a b) | HasInt b<br />
</haskell><br />
<br />
or a type function that substitutes one type with another inside<br />
arbitrarily-deep types:<br />
<br />
<haskell><br />
type Replace t a b | t==a = b<br />
Replace [t] a b = [Replace t a b]<br />
Replace (Set t) a b = Set (Replace t a b)<br />
Replace (Map t1 t2) a b = Map (Replace t1 a b) (Replace t2 a b)<br />
Replace t a b = t<br />
</haskell><br />
<br />
== One more hypothetical extension - multi-value type functions ==<br />
<br />
Let's add more fun! We will introduce one more hypothetical Haskell<br />
extension - type functions that may have MULTIPLE VALUES. Say,<br />
<br />
<haskell><br />
type Collection a = [a]<br />
Collection a = Set a<br />
Collection a = Map b a<br />
</haskell><br />
<br />
So, "Collection Int" has "[Int]", "Set Int" and "Map String Int" as<br />
its values, i.e. different collection types with elements of type<br />
"Int".<br />
<br />
Pay attention to the last statement of the "Collection" definition, where<br />
we used the type variable "b" that was not mentioned on the left side,<br />
nor defined in any other way. Since it's perfectly possible for the<br />
"Collection" function to have multiple values, using some free variable on<br />
the right side that can be replaced with any type is not a problem<br />
at all. "Map Bool Int", "Map [Int] Int" and "Map Int Int" all are<br />
possible values of "Collection Int" along with "[Int]" and "Set Int".<br />
<br />
At first glance, it seems that multiple-value functions are meaningless - they<br />
can't be used to define datatypes, because we need concrete types here. But<br />
if we take another look, they can be useful to define type constraints and<br />
type families.<br />
<br />
We can also represent a multiple-value function as a predicate:<br />
<br />
<haskell><br />
type Collection a [a]<br />
Collection a (Set a)<br />
Collection a (Map b a)<br />
</haskell><br />
<br />
If you're familiar with Prolog, you should know that a predicate, in contrast to<br />
a function, is a multi-directional thing - it can be used to deduce any<br />
parameter from the other ones. For example, in this hypothetical definition:<br />
<br />
<haskell><br />
head | Collection Int a :: a -> Int<br />
</haskell><br />
<br />
we define a 'head' function for any Collection containing Ints.<br />
<br />
And in this, again, hypothetical definition:<br />
<br />
<haskell><br />
data Safe c | Collection c a = Safe c a<br />
</haskell><br />
<br />
we deduced element type 'a' from collection type 'c' passed as the<br />
parameter to the type constructor.<br />
<br />
<br />
<br />
== Back to real Haskell - type classes ==<br />
<br />
After reading about all of these glorious examples, you may be wondering<br />
"Why doesn't Haskell support full-featured type functions?" Hold your breath...<br />
Haskell already contains them, and GHC has implemented all of the<br />
capabilities mentioned above for more than 10 years! They were just named...<br />
TYPE CLASSES! Let's translate all of our examples to their language:<br />
<br />
<haskell><br />
class IsSimple a<br />
instance IsSimple Bool<br />
instance IsSimple Int<br />
....<br />
instance IsSimple Double<br />
</haskell><br />
<br />
The Haskell'98 standard supports type classes with only one parameter.<br />
That limits us to only defining type predicates like this one. But GHC and<br />
Hugs support multi-parameter type classes that allow us to define<br />
arbitrarily-complex type functions<br />
<br />
<haskell><br />
class Collection a c<br />
instance Collection a [a]<br />
instance Collection a (Set a)<br />
instance Collection a (Map b a)<br />
</haskell><br />
<br />
All of the "hypothetical" Haskell extensions we investigated earlier are<br />
actually implemented at the type class level!<br />
<br />
Pattern matching:<br />
<br />
<haskell><br />
instance Collection a [a]<br />
</haskell><br />
<br />
Multiple statements:<br />
<br />
<haskell><br />
instance Collection a [a]<br />
instance Collection a (Set a)<br />
</haskell><br />
<br />
Recursion:<br />
<br />
<haskell><br />
instance (Collection a c) => Collection a [c]<br />
</haskell><br />
<br />
Pattern guards:<br />
<br />
<haskell><br />
instance (IsSimple a) => Collection a (UArray a)<br />
</haskell><br />
<br />
<br />
<br />
Let's define a type class which contains any collection which uses Int in<br />
its elements or indexes:<br />
<br />
<haskell><br />
class HasInt a<br />
instance HasInt Int<br />
instance (HasInt a) => HasInt [a]<br />
instance (HasInt a) => HasInt (Map a b)<br />
instance (HasInt b) => HasInt (Map a b)<br />
</haskell><br />
<br />
<br />
Another example is a class that replaces all occurrences of 'a' with<br />
'b' in type 't' and return the result as 'res':<br />
<br />
<haskell><br />
class Replace t a b res<br />
instance Replace t a a t<br />
instance Replace [t] a b [Replace t a b]<br />
instance (Replace t a b res)<br />
=> Replace (Set t) a b (Set res)<br />
instance (Replace t1 a b res1, Replace t2 a b res2)<br />
=> Replace (Map t1 t2) a b (Map res1 res2)<br />
instance Replace t a b t<br />
</haskell><br />
<br />
You can compare it to the hypothetical definition we gave earlier.<br />
It's important to note that type class instances, as opposed to<br />
function statements, are not checked in order. Instead, the most<br />
_specific_ instance is automatically selected. So, in the Replace case, the<br />
last instance, which is the most general instance, will be selected only if all the others<br />
fail to match, which is what we want.<br />
<br />
In many other cases this automatic selection is not powerful enough<br />
and we are forced to use some artificial tricks or complain to the<br />
language developers. The two most well-known language extensions<br />
proposed to solve such problems are instance priorities, which allow<br />
us to explicitly specify instance selection order, and '/=' constraints,<br />
which can be used to explicitly prohibit unwanted matches:<br />
<br />
<haskell><br />
instance Replace t a a t<br />
instance (a/=b) => Replace [t] a b [Replace t a b]<br />
instance (a/=b, t/=[_]) => Replace t a b t<br />
</haskell><br />
<br />
You can check that these instances no longer overlap.<br />
<br />
<br />
In practice, type-level arithmetic by itself is not very useful. It becomes a<br />
fantastic tool when combined with another feature that type classes provide -<br />
member functions. For example:<br />
<br />
<haskell><br />
class Collection a c where<br />
foldr1 :: (a -> a -> a) -> c -> a<br />
<br />
class Num a where<br />
(+) :: a -> a -> a<br />
<br />
sum :: (Num a, Collection a c) => c -> a<br />
sum = foldr1 (+)<br />
</haskell><br />
<br />
<br />
I'll also be glad to see the possibility of using type classes in data<br />
declarations, like this:<br />
<br />
<haskell><br />
data Safe c = (Collection c a) => Safe c a<br />
</haskell><br />
<br />
but as far as I know, this is not yet implemented.<br />
<br />
<br />
UNIFICATION<br />
...<br />
<br />
<br />
<br />
== Back to GADTs ==<br />
<br />
If you are wondering how all of these interesting type manipulations relate to<br />
GADTs, here is the answer. As you know, Haskell contains highly<br />
developed ways to express data-to-data functions. We also know that<br />
Haskell contains rich facilities to write type-to-type functions in the form of<br />
"type" statements and type classes. But how do "data" statements fit into this<br />
infrastructure?<br />
<br />
My answer: they just define a type-to-data constructor translation. Moreover,<br />
this translation may give multiple results. Say, the following definition:<br />
<br />
<haskell><br />
data Maybe a = Just a | Nothing<br />
</haskell><br />
<br />
defines type-to-data constructors function "Maybe" that has a parameter<br />
"a" and for each "a" has two possible results - "Just a" and<br />
"Nothing". We can rewrite it in the same hypothetical syntax that was<br />
used above for multi-value type functions:<br />
<br />
<haskell><br />
data Maybe a = Just a<br />
Maybe a = Nothing<br />
</haskell><br />
<br />
Or how about this:<br />
<br />
<haskell><br />
data List a = Cons a (List a)<br />
List a = Nil<br />
</haskell><br />
<br />
and this:<br />
<br />
<haskell><br />
data Either a b = Left a<br />
Either a b = Right b<br />
</haskell><br />
<br />
But how flexible are "data" definitions? As you should remember, "type"<br />
definitions were very limited in their features, while type classes,<br />
on the other hand, were more developed than ordinary Haskell functions<br />
facilities. What about features of "data" definitions examined as sort of functions?<br />
<br />
On the one hand, they supports multiple statements and multiple results and<br />
can be recursive, like the "List" definition above. On the other, that's all -<br />
no pattern matching or even type constants on the left side and no guards.<br />
<br />
Lack of pattern matching means that the left side can contain only free type<br />
variables. That in turn means that the left sides of all "data" statements for a<br />
type will be essentially the same. Therefore, repeated left sides in<br />
multi-statement "data" definitions are omitted and instead of<br />
<br />
<haskell><br />
data Either a b = Left a<br />
Either a b = Right b<br />
</haskell><br />
<br />
we write just<br />
<br />
<haskell><br />
data Either a b = Left a<br />
| Right b<br />
</haskell><br />
<br />
<br />
And here we finally come to GADTs! It's just a way to define data types using<br />
pattern matching and constants on the left side of "data" statements!<br />
Let's say we want to do this:<br />
<br />
<haskell><br />
data T String = D1 Int<br />
T Bool = D2<br />
T [a] = D3 (a,a)<br />
</haskell><br />
<br />
<br />
We cannot do this using a standard data definition. So, now we must use a GADT definition:<br />
<br />
<haskell><br />
data T a where<br />
D1 :: Int -> T String<br />
D2 :: T Bool<br />
D3 :: (a,a) -> T [a]<br />
</haskell><br />
<br />
Amazed? After all, GADTs seem to be a really simple and obvious extension to<br />
data type definition facilities.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
The idea here is to allow a data constructor's return type to be specified<br />
directly:<br />
<br />
<haskell><br />
data Term a where<br />
Lit :: Int -> Term Int<br />
Pair :: Term a -> Term b -> Term (a,b)<br />
...<br />
</haskell><br />
<br />
In a function that performs pattern matching on Term, the pattern match gives<br />
type as well as value information. For example, consider this function:<br />
<br />
<haskell><br />
eval :: Term a -> a<br />
eval (Lit i) = i<br />
eval (Pair a b) = (eval a, eval b)<br />
...<br />
</haskell><br />
<br />
If the argument matches Lit, it must have been built with a Lit constructor,<br />
so type 'a' must be Int, and hence we can return 'i' (an Int) in the right<br />
hand side. The same thing applies to the Pair constructor.<br />
<br />
<br />
<br />
<br />
<br />
<br />
== Further reading ==<br />
<br />
The best paper on type level arithmetic using type classes I've seen<br />
is "Faking it: simulating dependent types in Haskell"<br />
( http://www.cs.nott.ac.uk/~ctm/faking.ps.gz ). Most of<br />
this article comes from his work.<br />
<br />
A great demonstration of type-level arithmetic is in the TypeNats package,<br />
which "defines type-level natural numbers and arithmetic operations on<br />
them including addition, subtraction, multiplication, division and GCD"<br />
( darcs get --partial --tag '0.1' http://www.eecs.tufts.edu/~rdocki01/typenats/ )<br />
<br />
I should also mention here Oleg Kiselyov's page on type-level<br />
programming in Haskell: http://okmij.org/ftp/Haskell/types.html<br />
<br />
<br />
There are plenty of GADT-related papers, but the best one for beginners<br />
is "Fun with phantom types"<br />
(http://www.informatik.uni-bonn.de/~ralf/publications/With.pdf).<br />
Phantom types is another name of GADT. You should also know that this<br />
paper uses old GADT syntax. This paper is a must-read because it<br />
contains numerous examples of practical GADT usage - a theme completely<br />
omitted from my article.<br />
<br />
<br />
Other GADT-related papers:<br />
<br />
"Dynamic Optimization for Functional Reactive Programming using<br />
Generalized Algebraic Data Types"<br />
http://www.cs.nott.ac.uk/~nhn/Publications/icfp2005.pdf<br />
<br />
"Phantom types" (actually more scientific version of "Fun with phantom types")<br />
http://citeseer.ist.psu.edu/rd/0,606209,1,0.25,Download/http:qSqqSqwww.informatik.uni-bonn.deqSq~ralfqSqpublicationsqSqPhantom.ps.gz<br />
<br />
"Phantom types and subtyping" http://arxiv.org/ps/cs.PL/0403034<br />
<br />
"Existentially quantified type classes" by Stuckey, Sulzmann and Wazny (URL?)</div>Blackouthttps://wiki.haskell.org/index.php?title=Comparison_of_functional_programming_languages&diff=55401Comparison of functional programming languages2013-02-07T18:50:33Z<p>Blackout: w/ -> with. We don't need to save bandwidth.</p>
<hr />
<div>[[Category:Glossary]]<br />
<br />
{| class="wikitable" style="text-align:center;"<br />
|<br />
! Evaluation<br />
! IO<br />
! Typing<br />
! Pure<br />
|-<br />
! Haskell<br />
| lazy<br />
| monads<br />
| static<br />
| yes<br />
|-<br />
! Clean<br />
| lazy<br />
| uniqueness<br />
| static<br />
| yes<br />
|-<br />
! Miranda<br />
| lazy<br />
| lazy lists<br />
| static<br />
| yes<br />
|-<br />
! ML<br />
| eager<br />
| side-effects<br />
| static<br />
| no<br />
|-<br />
! Scheme<br />
| eager<br />
| side-effects<br />
| dynamic<br />
| no<br />
|-<br />
! Erlang<br />
| eager<br />
| side-effects<br />
| dynamic with compiler annotations for static typing<br />
| no<br />
|}</div>Blackouthttps://wiki.haskell.org/index.php?title=Tying_the_Knot&diff=55372Tying the Knot2013-02-03T12:48:12Z<p>Blackout: markup</p>
<hr />
<div>In a language like Haskell, where Lists are defined as <hask>Nil | Cons a (List a)</hask>, creating data structures like cyclic or doubly linked lists seems impossible. However, this is not the case: laziness allows for such definitions, and the procedure of doing so is called ''tying the knot''. The simplest example:<br />
<br />
<haskell><br />
cyclic = let x = 0 : y<br />
y = 1 : x<br />
in x<br />
</haskell><br />
<br />
This creates the cyclic list consisting of 0 and 1. It is important to stress that this procedure allocates only two numbers - 0 and 1 - in memory, making this a truly cyclic list.<br />
<br />
The knot analogy stems from the fact that we produce two open-ended objects, and then link their ends together. Evaluation of the above therefore looks something like<br />
<br />
<haskell><br />
cyclic<br />
= x<br />
= 0 : y<br />
= 0 : 1 : x -- Knot! Back to the beginning.<br />
= 0 : 1 : 0 : x<br />
= -- etc.<br />
</haskell><br />
<br />
<br />
== Overview ==<br />
<br />
This example illustrates different ways to define recursive data structures.<br />
To demonstrate the different techniques we show how to solve the same problem---writing an interpreter for a simple programming language---in three different ways. This is a nice example because, (i) it is interesting, (ii) the abstract syntax of the language contains mutually recursive structures, and (iii) the interpreter illustrates how to work with the recursive structures.<br />
<br />
(It would be useful to have some more text describing the examples.)<br />
<br />
== Download the files ==<br />
* [[Media:Interp1.lhs|Direct Recursion]]<br />
* [[Media:Interp2.lhs|Tying the Knot]]<br />
* [[Media:Interp3.lhs|Tying the Knot with GADTs]]<br />
<br />
== Other Examples ==<br />
<br />
* [http://twan.home.fmf.nl/blog/haskell/Knuth-Morris-Pratt-in-Haskell.details Knuth-Morris-Pratt algorithm for substring matching]<br />
<br />
== Migrated from the old wiki ==<br />
<br />
How to build a cyclic data structure.<br />
<br />
At first, the lack of pointer-updating operations in Haskell makes it seem that building cyclic structures (circular lists, graphs, etc) is impossible. This is not the case, thanks to the ability to define data using recursive equations. Here is a little trick called `tying the knot'.<br />
<br />
Remember that Haskell is a lazy language. A consequence of this is that while you are building the node, you can set the children to the final values straight away, even though you don't know them yet! It twists your brain a bit the first few times you do it, but it works fine.<br />
<br />
Here's an example. Say you want to build a circular, doubly-linked list, given a standard Haskell list as input. The back pointers are easy, but what about the forward ones?<br />
<haskell><br />
data DList a = DLNode (DList a) a (DList a)<br />
<br />
mkDList :: [a] -> DList a<br />
<br />
mkDList [] = error "must have at least one element"<br />
mkDList xs = let (first,last) = go last xs first<br />
in first<br />
<br />
where go :: DList a -> [a] -> DList a -> (DList a, DList a)<br />
go prev [] next = (next,prev)<br />
go prev (x:xs) next = let this = DLNode prev x rest<br />
(rest,last) = go this xs next<br />
in (this,last)<br />
<br />
takeF :: Integer -> DList a -> [a]<br />
takeF 0 _ = []<br />
takeF (n+1) (DLNode _ x next) = x : (takeF n next)<br />
<br />
takeR :: Show a => Integer -> DList a -> [a]<br />
takeR 0 _ = []<br />
takeR (n+1) (DLNode prev x _) = x : (takeR n prev)<br />
</haskell><br />
(takeF and takeR are simply to let you look at the results of mkDList: they take a specified number of elements, either forward or backward).<br />
<br />
The trickery takes place in `go'. `go' builds a segment of the list, given a pointer to the node off to the left of the segment and off to the right. Look at the second case of `go'. We build the first node of the segment, using the given prev pointer for the left link, and the node pointer we are *about* to compute in the next step for the right link.<br />
<br />
This goes on right the way through the segment. But how do we manage to create a *circular* list this way? How can we know right at the beginning what the pointer to the end of the list will be?<br />
<br />
Take a look at mkDList. Here, we simply take the (first,last) pointers we get from `go', and *pass them back in* as the next and prev pointers respectively, thus tying the knot. This all works because of lazy evaluation.<br />
<br />
Somehow the following seems more straightforward to me, though perhaps I'm missing the point here:<br />
<haskell><br />
data DList a = DLNode (DList a) a (DList a)<br />
<br />
rot :: Integer -> [a] -> [a]<br />
rot n xs | n < 0 = rot (n+1) ((last xs):(init xs))<br />
| n == 0 = xs<br />
| n > 0 = rot (n-1) (tail xs ++ [head xs])<br />
<br />
mkDList :: [a] -> DList a<br />
mkDList [] = error "Must have at least one element."<br />
mkDList xs = DLNode (mkDList $ rot (-1) xs) (head xs) (mkDList $ rot 1 xs)<br />
</haskell><br />
<br />
- CaleGibbard<br />
<br />
The problem with this is it won't make a truly cyclic data structure, rather it will constantly be generating the rest of the list. To see this use trace (in Debug.Trace for GHC) in mkDList (e.g. mkDList xs = trace "mkDList" $ ...) and then takeF 10 (mkDList "a"). Add a trace to mkDList or go or wherever you like in the other version and note the difference.<br />
<br />
-- DerekElkins<br />
<br />
Yeah, thanks, I see what you mean.<br />
<br />
This is so amazing that everybody should have seen it, so here's the trace. I put trace "\n--go{1/2}--" $ directly after the two go definitions:<br />
<br />
<haskell><br />
*Main> takeF 10 $ mkDList [1..3]<br />
<br />
--go2--<br />
[1<br />
--go2--<br />
,2<br />
--go2--<br />
,3<br />
--go1--<br />
,1,2,3,1,2,3,1]<br />
<br />
</haskell><br />
<br />
- CaleGibbard<br />
<br />
The above works for simple cases, but sometimes you need construct some very complex data structures, where the pattern of recursion is not known at compile time. If this is the case, may need to use an auxiliary dictionary data structure to help you tie your knots.<br />
<br />
Consider, for example, how you would implement deterministic finite automata (DFAs). One possibility is:<br />
<haskell><br />
type IndirectDfa a = (Int, [IndirectState a])<br />
<br />
data IndirectState a =<br />
IndirectState Bool [(a, Int)]<br />
</haskell><br />
That is, a DFA is a set of states, one of which is distinguished as being the "start state". Each state has a number of transitions which lead to other states, as well as a flag which specifies or not the state is final.<br />
<br />
This representation is fine for manipulation, but it's not as suitable for actually executing the DFA as it could be because we need to "look up" a state every time we make a transition. There are relatively cheap ways to implement this indirection, of course, but ideally we shouldn't have to pay much for it.<br />
<br />
What we really want is a recursive data structure:<br />
<br />
<haskell><br />
data DirectDfa a<br />
= DirectState Bool [(a, DirectDfa a)]<br />
</haskell><br />
Then we can just execute the DFA like this:<br />
<haskell><br />
runDfa :: (Eq a) => DirectDfa a -> [a] -> Bool<br />
runDfa (DirectState final trans) []<br />
= final<br />
runDfa (DirectState final trans) (x:xs)<br />
= case [ s | (x',s) <- trans, x == x' ] of<br />
[] -> False<br />
(s:_) -> runDfa s xs<br />
</haskell><br />
(Note: We're only optimising state lookup here, not deciding which transition to take. As an exercise, consider how you might optimise transitions. You may wish to use [[RunTimeCompilation]].)<br />
<br />
Turning the indirect recursion into direct recursion requires tying knots, and it's not immediately obvious how to do this by holding onto lazy pointers, because any state can potentially point to any other state (or, indeed, every other state).<br />
<br />
What we can do is introduce a dictionary data structure to hold the (lazily evaluated) new states, then introducing a recursive reference can be done with a a simple dictionary lookup. In principle, you could use any dictionary data structure (e.g. Map). However, in this case, the state numbers are dense integers, so it's probably easiest to use an array:<br />
<haskell><br />
indirectToDirect :: IndirectDfa a -> DirectDfa a<br />
indirectToDirect (start, states)<br />
= tieArray ! start<br />
where<br />
tieArray = array (0,length states - 1)<br />
[ (i,direct s) | (i,s) <- zip [0..] states ]<br />
<br />
direct (IndirectState final trans)<br />
= DirectState final [ (x, tieArray ! s) | (x,s) <- trans ]<br />
</haskell><br />
Note how similar this is to the technique of [[MemoisingCafs]]. In fact what we've done here is "memoised" the data structure, using something like [[HashConsing]].<br />
<br />
As noted previously, pretty much any dictionary data structure will do. Often you can even use structures with slow lookup (e.g. association lists). This is because fully lazy evaluation ensures that you only pay for each lookup the first time you use it; subsequent uses of the same part of the data structure are effectively free.<br />
<br />
-- AndrewBromage<br />
<br />
Transformations of cyclic graphs and the Credit Card Transform<br />
<br />
Cycles certainly make it difficult to transform graphs in a pure non-strict language. Cycles in a source graph require us to devise a way to mark traversed nodes -- however we cannot mutate nodes and cannot even compare nodes with a generic ('derived') equality operator. Cycles in a destination graph require us to keep track of the already constructed nodes so we can complete a cycle. An obvious solution is to use a state monad and IORefs. There is also a monad-less solution, which is less obvious: seemingly we cannot add a node to the dictionary of already constructed nodes until we have built the node. This fact means that we cannot use the updated dictionary when building the descendants of the node -- which need the updated dictionary to link back. The problem can be overcome however with a ''credit card transform'' (a.k.a. "buy now, pay later" transform). To avoid hitting the bottom, we just have to "pay" by the "due date".<br />
<br />
For illustration, we will consider the problem of printing out a non-deterministic finite automaton (NFA) and transforming it into a deterministic finite automaton (DFA). Both NFA and DFA are represented as cyclic graphs. The problem has been discussed on the Haskell/Haskell-Cafe mailing lists. The automata in question were to recognize strings over a binary alphabet.<br />
<br />
A state of an automaton over a binary alphabet is a data structure:<br />
<haskell><br />
data (Ord l,Show l) => FaState l =<br />
FaState {label :: l, acceptQ :: Bool,<br />
trans0:: [FaState l],<br />
trans1:: [FaState l]}<br />
</haskell><br />
whose fields have the obvious meaning. Label is used for printing out and comparing states. The flag acceptQ tells if the state is final. Since an FaState can generally represent a non-deterministic automaton, transitions are the ''lists'' of states.<br />
<br />
An automaton is then a list of starting states.<br />
<haskell><br />
type FinAu l = [FaState l]<br />
</haskell><br />
For example, an automaton equivalent to the regular expression "0*(0(0+1)*)*" could be defined as:<br />
<haskell><br />
dom18 = [one]<br />
where one = FaState 1 True [one,two] []<br />
two = FaState 2 True [two,one] [one,two]<br />
</haskell><br />
using the straightforward translation from a regular expression to an NFA.<br />
<br />
We would like to compare and print automata and their states:<br />
<haskell><br />
instance (Ord l,Show l) => Eq (FaState l) where<br />
(FaState l1 _ _ _) == (FaState l2 _ _ _) = l1 == l2<br />
</haskell><br />
<br />
Printing a FaState however poses a slight problem. For example, the state labeled '1' in the automaton dom18 refers to itself. If we blindly 'follow the links', we will loop forever. Therefore, we must keep track of the already printed states. We need a data structure for such an occurrence check, with the following obvious operations:<br />
<br />
<haskell><br />
class OCC occ where<br />
empty:: occ a<br />
seenp:: (Eq a) => a -> occ a -> Bool -- occurrence check predicate<br />
put:: a -> occ a -> occ a -- add an item<br />
</haskell><br />
<br />
In this article, we realize such a data structure as a list. In the future, we can pull in something fancier from the Edison collection:<br />
<br />
<haskell><br />
instance OCC [] where<br />
empty = []<br />
seenp = elem<br />
put = (:)<br />
</haskell><br />
<br />
We are now ready to print an automaton. To be more precise, we traverse the corresponding graph depth-first, pre-order, and keep track of the already printed states. A 'states_seen' datum accumulates the shown states, so we can be sure we print each state only once and thus avoid the looping.<br />
<br />
<haskell><br />
instance (Ord l,Show l) => Show (FaState l) where<br />
show state = "{@" ++ showstates [state] (empty::[FaState l]) "@}"<br />
where<br />
-- showstates worklist seen_states suffix<br />
showstates [] states_seen suffix = suffix<br />
showstates (st:rest) states_seen suffix<br />
| st `seenp` states_seen = showstates rest states_seen suffix<br />
showstates (st@(FaState l accept t0 t1):rest) states_seen suffix =<br />
showstate st<br />
$ showstates (t0++t1++rest) (st `put` states_seen) suffix<br />
<br />
showstate (FaState l accept t0 t1) suffix<br />
= "{State " ++ (show l) ++<br />
" " ++ (show accept) ++ " " ++ (show $ map label t0) ++<br />
" " ++ (show $ map label t1) ++ "}" ++ suffix<br />
</haskell><br />
<br />
Now,<br />
<br />
<haskell><br />
CCardFA> print dom18 -- prints as<br />
CCardFA> [{@{State 1 True [1,2] []}{State 2 True [2,1] [1,2]}@}]<br />
</haskell><br />
<br />
The acceptance function for our automata can be written as follows. The function takes the list of starting states and the string over the boolean alphabet. The function returns True if the string is accepted.<br />
<br />
<haskell><br />
finAuAcceptStringQ start_states str =<br />
any (\l -> acceptP l str) start_states<br />
where acceptP (FaState _ acceptQ _ _) [] = acceptQ<br />
acceptP (FaState _ _ t0 t1) (s:rest) =<br />
finAuAcceptStringQ (if s then t1 else t0) rest<br />
</haskell><br />
<br />
To test the automata, we can try<br />
<br />
<haskell><br />
test1= finAuAcceptStringQ dom18 $ map (>0) [0,1,0,1]<br />
test2= finAuAcceptStringQ dom18 $ map (>0) [1,1,0,1]<br />
test3= finAuAcceptStringQ dom18 [True]<br />
test4= finAuAcceptStringQ dom18 [False]<br />
</haskell><br />
<br />
We are now ready to write the NFA->DFA conversion, a determinization of an NFA. We implement the textbook algorithm of tracing set of NFA states. A state in the resulting DFA corresponds to a list of the NFA states. A DFA is generally a cyclic graph, often with cycles of length 1 (self-referenced nodes). To be able to "link back" as we build DFA states, we have to remember the already constructed states. We need a data structure, a dictionary of states:<br />
<br />
<haskell><br />
class StateDict sd where<br />
emptyd :: sd (l,FaState l)<br />
locate :: (Eq l) => l -> sd (l,FaState l) -> Maybe (FaState l)<br />
putd :: (l,FaState l) -> sd (l,FaState l) -> sd (l,FaState l)<br />
</haskell><br />
<br />
For now, we realize this dictionary as an associative list. If performance matters, we can use a fancier dictionary from the Edison<br />
<br />
<haskell><br />
instance StateDict [] where<br />
emptyd = []<br />
locate = lookup<br />
putd = (:)<br />
</haskell><br />
<br />
The work of the NFA->DFA conversion is done by the following function determinize_cc. The function takes a list of NFA states, the dictionary of the already built states, and returns a pair <hask>([dfa_state], updated_dictionary)</hask> where <hask>[dfa_state]</hask> is a singleton list.<br />
<br />
<haskell><br />
-- [nfa_state] -> dictionary_of_seen_states -><br />
-- ([dfa_state],updated_dictionary)<br />
-- [dfa_state] is a singleton list<br />
determinize_cc states converted_states =<br />
-- first, check the cache to see if the state has been built already<br />
case dfa_label `locate` converted_states of<br />
Nothing -> build_state<br />
Just dfa_state -> ([dfa_state],converted_states)<br />
where<br />
-- [NFA_labels] -> DFA_labels<br />
det_labels = sort . nub . map label<br />
dfa_label = det_labels states<br />
<br />
-- find out NFA-followers for [nfa_state] upon ingestion of 0 and 1<br />
(t0_followers,t1_followers) =<br />
foldr (\st (f0,f1) -> (trans0 st ++ f0, trans1 st ++ f1))<br />
([],[]) states<br />
acceptQ' = any acceptQ states<br />
<br />
-- really build the dfa state and return ([dfa_state],updated_cache)<br />
build_state = let<br />
-- note, the dfa_state is computed _below_<br />
converted_states1 = (dfa_label,dfa_state) `putd` converted_states<br />
(t0', converted_states2) =<br />
(determinize_cc t0_followers converted_states1)<br />
(t1', converted_states3) =<br />
(determinize_cc t1_followers converted_states2)<br />
dfa_state =<br />
(FaState dfa_label acceptQ' t0' t1')<br />
in ([dfa_state],converted_states3)<br />
</haskell><br />
The front end of the NFA->DFA transformer:<br />
<hask>finAuDeterminize states = fst $ determinize_cc states []</hask><br />
<br />
At the heart of the credit card transform is the phrase from the above code:<br />
<hask> converted_states1 = (dfa_label,dfa_state) `putd` converted_states</hask><br />
<br />
The phrase expresses the addition to the dictionary of the 'converted_states' of a 'dfa_state' that we haven't built yet. The computation of the 'dfa_state' is written 4 lines below the phrase in question. Because (,) is non-strict in its arguments and locate is non-strict in its result, we can get away with a mere promise to "pay". Note that the computation of the dfa_state needs t0' and t1', which in turn rely on 'converted_states1'. This fact shows that we can tie the knot by making a promise to compute a state, add this promise to the dictionary of the built states, and use the updated dictionary to build the descendants. Because Haskell is a non-strict language, we don't need to do anything special to make the promise. Every computation is Haskell is by default a promise.<br />
<br />
We can print the DFA for dom18 to see what we've got:<br />
<haskell><br />
CCardFA> finAuDeterminize dom18<br />
CCardFA>-- which shows<br />
CCardFA> [{@{State [1] True [[1,2]] [[]] }<br />
CCardFA> {State [1,2] True [[1,2]] [[1,2]]}<br />
CCardFA> {State [] False [[]] [[]] }@}]<br />
</haskell><br />
which is indeed a DFA (which happens to be minimal) recognizing (0+1)* - 1(0+1)*<br />
<br />
We can run the determinized FA using the same function finAuAcceptStringQ:<br />
<haskell><br />
test1' = finAuAcceptStringQ (finAuDeterminize dom18) $ map (>0) [0,1,0,1]<br />
test2' = finAuAcceptStringQ (finAuDeterminize dom18) $ map (>0) [1,1,0,1]<br />
</haskell><br />
<br />
The complete code for this example is in http://pobox.com/~oleg/ftp/Haskell/CCard-transform-DFA.lhs. Another example of tying a knot in the case of forward links, by using a fixed-point combinator, is discussed in http://www.mail-archive.com/haskell@haskell.org/msg10687.html.<br />
<br />
-- Oleg<br />
<br />
For a long time, I've had an issue with Oleg's reply to Hal Daume III, the "forward links" example. The problem is that it doesn't really exploit laziness or circular values. It's solution would work even in a strict language. It's simply a functional version of the standard approach: build the result with markers and patch it up afterwards. It is a fairly clever way of doing purely something that is typically done with references and mutable update, but it doesn't really address what Hal Daume III was after. Fixing Hal Daume's example so that it won't loop is relatively trivial; simply change the case to a let or equivalently use a lazy pattern match in the case. However, if that's all there was to it, I would've written this a long time ago.<br />
<br />
The problem is that it no longer gives you control of the error message or anyway to recover from it. With GHC's extensions to exception handling you could do it, but you'd have to put readDecisionTree in the IO monad to recover from it, and if you wanted better messages you'd have to put most of the parsing in the IO monad so that you could catch the error earlier and provide more information then rethrow. What's kept me is that I couldn't figure out a way to tie the knot when the environment had a type like, Either String [(String,DecisionTree)]. This is because it's impossible for this case; we decide whether to return Left "could not find subtree" or Right someValue and therefore whether the environment is Left or Right based on whether we could find the subtree in the environment. In effect, we need to lookup a value in an environment we may return to know whether to return it. Obviously this is a truly circular dependency. This made me think that Oleg's solution was as good as any other and better than some (actually, ironically Oleg's solution also uses a let instead of a case, however, there's nothing stopping it from being a case, but it still would provide no way to recover from it without effectively doing what is mentioned below). Recently, I've thought about this again and the solution is obvious and follows directly from the original definition modified to use let. It doesn't loop because only particular values in the lookup table fail, in fact, you might never know there was a variable lookup error if you didn't touch all of the tree. This translates directly into the environment having type [(String,Either String DecisionTree)]. There are several benefits to this approach compared to Oleg's: 1) it solves my original problem, you are now able to specify the error messages (Oleg's can do this), 2) it goes beyond that (and beyond Hal Daume's original "specification") and also allows you to recover from an error without resorting to the IO monad and/or extensions (Oleg's can't do this), 3) it does implicitly what Oleg's version does explicitly, 4) because of (3) it shares properly while Oleg's does not*, 5) both the environment and the returned value are made up of showable values, not opaque functions, 6) it requires less changes to the original code and is more localized than Oleg's solution; only the variable lookup and top-level function will need to change.<br />
<br />
To recover, all one needs to do is make sure all the values in the lookup table are Right values. If they aren't, there are various ways you could collect the information; there are also variations on how to combine error information and what to provide. Even without a correctness check, you can still provide better error messages for the erroneous thunks. A possible variation that loses some of the benefits, is to change the DecisionTree type (or have a different version, [[IndirectComposite]] comes to mind here) that has Either ErrorInfo ErrorDecisionTree subnodes, which will allow you to recover at any time (though, if you want to make a normal DecisionTree out of it you will lose sharing). Also, the circular dependency only comes up if you need to use the environment to decide on an error. For example, a plain old syntactic parse error can cyclicly use an Either ErrorInfo [(String,DecisionTree)] perfectly fine (pass in fromRight env where fromRight ~(Right x) = x). It will also work even with the above approach giving the environment the type Either [(String,Either ErrorInfo DecisionTree)]. Below is code for a simplified scenario that does most of these things,<br />
<br />
<haskell><br />
module Main where<br />
import Maybe ( fromJust )<br />
import Monad<br />
<br />
main :: IO ()<br />
main = do<br />
input <- getContents<br />
length input `seq` print (fixup input)<br />
<br />
instance Monad (Either s) where<br />
return = Right<br />
m >>= f = either Left f m<br />
<br />
isLeft :: Either a b -> Bool<br />
isLeft (Left _) = True<br />
isLeft _ = False<br />
<br />
fromRight :: Either a b -> b<br />
fromRight ~(Right x) = x<br />
<br />
fixup :: String -> Either String [(String,Either String String)]<br />
fixup input = env<br />
where env = mapM (parse (fromRight env) . words) (lines input)<br />
<br />
checkedFixup :: String -> Either String [(String,String)]<br />
checkedFixup input =<br />
case fixup input of<br />
Left err -> Left err<br />
Right env -> case filter (isLeft . snd) env of<br />
[] -> Right $ map (\(n,Right v) -> (n,v)) env<br />
(_,Left err):_ -> Left err<br />
<br />
parse :: [(String,Either String String)] -> [String] -> Either String (String,Either String String)<br />
parse env ("define":name:values) = Right (name,values')<br />
where values' = liftM unwords $ mapM lookupRef values<br />
lookupRef ('*':word)<br />
= maybe (Left $ "couldn't find "++word++" in "++name)<br />
id<br />
(lookup word env)<br />
lookupRef word = Right word<br />
parse env input = Left $ "parse error with: "++unwords input<br />
</haskell><br />
checkedFixup demonstrates how you could check and recover, but since the environment is the return value neither fixup or checkedFixup quite illustrate having potentially erroneous thunks in the actual return value. Some example input,<br />
<haskell><br />
define x *y *y<br />
define y a b<br />
outputs Right [("x",Right "a b a b"),("y",Right "a b")]<br />
define x *y *y<br />
aousht<br />
define y a b<br />
outputs Left "parse error with: aousht"<br />
define x *y *z<br />
define y a b<br />
define z *w<br />
outputs Right [("x",Left "couldn't find w in z"),("y",Right "a b"),("z",Left "couldn't find w in z")]<br />
</haskell><br />
<br />
* Consider a tree Y that contains the subtree X twice. With Oleg's version, when we resolve the "X" variable we look up a (manually) delayed tree and then build X. Each subtree of Y will build it's own version of X. With the truly circular version each subtree of Y will be the same, possibly erroneous, thunk that builds X, if the thunk isn't erroneous then when it is updated both of Y's subtrees will point to the same X.<br />
<br />
-- DerekElkins<br />
<br />
<br />
[[Category:Code]]<br />
[[Category:Idioms]]</div>Blackouthttps://wiki.haskell.org/index.php?title=Tying_the_Knot&diff=55371Tying the Knot2013-02-03T12:46:15Z<p>Blackout: Introductory statement added</p>
<hr />
<div>In a language like Haskell, where Lists are defined as <hask>Nil | Cons a (List a)</hask>, creating data structures like cyclic or doubly linked lists seems impossible. However, this is not the case: laziness allows for such definitions, and the procedure of doing so is called *tying the knot*. The simplest example:<br />
<br />
<haskell><br />
cyclic = let x = 0 : y<br />
y = 1 : x<br />
in x<br />
</haskell><br />
<br />
This creates the cyclic list consisting of 0 and 1. It is important to stress that this procedure allocates only two numbers - 0 and 1 - in memory, making this a truly cyclic list.<br />
<br />
The knot analogy stems from the fact that we produce two open-ended objects, and then link their ends together. Evaluation of the above therefore looks something like<br />
<br />
<haskell><br />
cyclic<br />
= x<br />
= 0 : y<br />
= 0 : 1 : x -- Knot! Back to the beginning.<br />
= 0 : 1 : 0 : x<br />
= -- etc.<br />
</haskell><br />
<br />
<br />
== Overview ==<br />
<br />
This example illustrates different ways to define recursive data structures.<br />
To demonstrate the different techniques we show how to solve the same problem---writing an interpreter for a simple programming language---in three different ways. This is a nice example because, (i) it is interesting, (ii) the abstract syntax of the language contains mutually recursive structures, and (iii) the interpreter illustrates how to work with the recursive structures.<br />
<br />
(It would be useful to have some more text describing the examples.)<br />
<br />
== Download the files ==<br />
* [[Media:Interp1.lhs|Direct Recursion]]<br />
* [[Media:Interp2.lhs|Tying the Knot]]<br />
* [[Media:Interp3.lhs|Tying the Knot with GADTs]]<br />
<br />
== Other Examples ==<br />
<br />
* [http://twan.home.fmf.nl/blog/haskell/Knuth-Morris-Pratt-in-Haskell.details Knuth-Morris-Pratt algorithm for substring matching]<br />
<br />
== Migrated from the old wiki ==<br />
<br />
How to build a cyclic data structure.<br />
<br />
At first, the lack of pointer-updating operations in Haskell makes it seem that building cyclic structures (circular lists, graphs, etc) is impossible. This is not the case, thanks to the ability to define data using recursive equations. Here is a little trick called `tying the knot'.<br />
<br />
Remember that Haskell is a lazy language. A consequence of this is that while you are building the node, you can set the children to the final values straight away, even though you don't know them yet! It twists your brain a bit the first few times you do it, but it works fine.<br />
<br />
Here's an example. Say you want to build a circular, doubly-linked list, given a standard Haskell list as input. The back pointers are easy, but what about the forward ones?<br />
<haskell><br />
data DList a = DLNode (DList a) a (DList a)<br />
<br />
mkDList :: [a] -> DList a<br />
<br />
mkDList [] = error "must have at least one element"<br />
mkDList xs = let (first,last) = go last xs first<br />
in first<br />
<br />
where go :: DList a -> [a] -> DList a -> (DList a, DList a)<br />
go prev [] next = (next,prev)<br />
go prev (x:xs) next = let this = DLNode prev x rest<br />
(rest,last) = go this xs next<br />
in (this,last)<br />
<br />
takeF :: Integer -> DList a -> [a]<br />
takeF 0 _ = []<br />
takeF (n+1) (DLNode _ x next) = x : (takeF n next)<br />
<br />
takeR :: Show a => Integer -> DList a -> [a]<br />
takeR 0 _ = []<br />
takeR (n+1) (DLNode prev x _) = x : (takeR n prev)<br />
</haskell><br />
(takeF and takeR are simply to let you look at the results of mkDList: they take a specified number of elements, either forward or backward).<br />
<br />
The trickery takes place in `go'. `go' builds a segment of the list, given a pointer to the node off to the left of the segment and off to the right. Look at the second case of `go'. We build the first node of the segment, using the given prev pointer for the left link, and the node pointer we are *about* to compute in the next step for the right link.<br />
<br />
This goes on right the way through the segment. But how do we manage to create a *circular* list this way? How can we know right at the beginning what the pointer to the end of the list will be?<br />
<br />
Take a look at mkDList. Here, we simply take the (first,last) pointers we get from `go', and *pass them back in* as the next and prev pointers respectively, thus tying the knot. This all works because of lazy evaluation.<br />
<br />
Somehow the following seems more straightforward to me, though perhaps I'm missing the point here:<br />
<haskell><br />
data DList a = DLNode (DList a) a (DList a)<br />
<br />
rot :: Integer -> [a] -> [a]<br />
rot n xs | n < 0 = rot (n+1) ((last xs):(init xs))<br />
| n == 0 = xs<br />
| n > 0 = rot (n-1) (tail xs ++ [head xs])<br />
<br />
mkDList :: [a] -> DList a<br />
mkDList [] = error "Must have at least one element."<br />
mkDList xs = DLNode (mkDList $ rot (-1) xs) (head xs) (mkDList $ rot 1 xs)<br />
</haskell><br />
<br />
- CaleGibbard<br />
<br />
The problem with this is it won't make a truly cyclic data structure, rather it will constantly be generating the rest of the list. To see this use trace (in Debug.Trace for GHC) in mkDList (e.g. mkDList xs = trace "mkDList" $ ...) and then takeF 10 (mkDList "a"). Add a trace to mkDList or go or wherever you like in the other version and note the difference.<br />
<br />
-- DerekElkins<br />
<br />
Yeah, thanks, I see what you mean.<br />
<br />
This is so amazing that everybody should have seen it, so here's the trace. I put trace "\n--go{1/2}--" $ directly after the two go definitions:<br />
<br />
<haskell><br />
*Main> takeF 10 $ mkDList [1..3]<br />
<br />
--go2--<br />
[1<br />
--go2--<br />
,2<br />
--go2--<br />
,3<br />
--go1--<br />
,1,2,3,1,2,3,1]<br />
<br />
</haskell><br />
<br />
- CaleGibbard<br />
<br />
The above works for simple cases, but sometimes you need construct some very complex data structures, where the pattern of recursion is not known at compile time. If this is the case, may need to use an auxiliary dictionary data structure to help you tie your knots.<br />
<br />
Consider, for example, how you would implement deterministic finite automata (DFAs). One possibility is:<br />
<haskell><br />
type IndirectDfa a = (Int, [IndirectState a])<br />
<br />
data IndirectState a =<br />
IndirectState Bool [(a, Int)]<br />
</haskell><br />
That is, a DFA is a set of states, one of which is distinguished as being the "start state". Each state has a number of transitions which lead to other states, as well as a flag which specifies or not the state is final.<br />
<br />
This representation is fine for manipulation, but it's not as suitable for actually executing the DFA as it could be because we need to "look up" a state every time we make a transition. There are relatively cheap ways to implement this indirection, of course, but ideally we shouldn't have to pay much for it.<br />
<br />
What we really want is a recursive data structure:<br />
<br />
<haskell><br />
data DirectDfa a<br />
= DirectState Bool [(a, DirectDfa a)]<br />
</haskell><br />
Then we can just execute the DFA like this:<br />
<haskell><br />
runDfa :: (Eq a) => DirectDfa a -> [a] -> Bool<br />
runDfa (DirectState final trans) []<br />
= final<br />
runDfa (DirectState final trans) (x:xs)<br />
= case [ s | (x',s) <- trans, x == x' ] of<br />
[] -> False<br />
(s:_) -> runDfa s xs<br />
</haskell><br />
(Note: We're only optimising state lookup here, not deciding which transition to take. As an exercise, consider how you might optimise transitions. You may wish to use [[RunTimeCompilation]].)<br />
<br />
Turning the indirect recursion into direct recursion requires tying knots, and it's not immediately obvious how to do this by holding onto lazy pointers, because any state can potentially point to any other state (or, indeed, every other state).<br />
<br />
What we can do is introduce a dictionary data structure to hold the (lazily evaluated) new states, then introducing a recursive reference can be done with a a simple dictionary lookup. In principle, you could use any dictionary data structure (e.g. Map). However, in this case, the state numbers are dense integers, so it's probably easiest to use an array:<br />
<haskell><br />
indirectToDirect :: IndirectDfa a -> DirectDfa a<br />
indirectToDirect (start, states)<br />
= tieArray ! start<br />
where<br />
tieArray = array (0,length states - 1)<br />
[ (i,direct s) | (i,s) <- zip [0..] states ]<br />
<br />
direct (IndirectState final trans)<br />
= DirectState final [ (x, tieArray ! s) | (x,s) <- trans ]<br />
</haskell><br />
Note how similar this is to the technique of [[MemoisingCafs]]. In fact what we've done here is "memoised" the data structure, using something like [[HashConsing]].<br />
<br />
As noted previously, pretty much any dictionary data structure will do. Often you can even use structures with slow lookup (e.g. association lists). This is because fully lazy evaluation ensures that you only pay for each lookup the first time you use it; subsequent uses of the same part of the data structure are effectively free.<br />
<br />
-- AndrewBromage<br />
<br />
Transformations of cyclic graphs and the Credit Card Transform<br />
<br />
Cycles certainly make it difficult to transform graphs in a pure non-strict language. Cycles in a source graph require us to devise a way to mark traversed nodes -- however we cannot mutate nodes and cannot even compare nodes with a generic ('derived') equality operator. Cycles in a destination graph require us to keep track of the already constructed nodes so we can complete a cycle. An obvious solution is to use a state monad and IORefs. There is also a monad-less solution, which is less obvious: seemingly we cannot add a node to the dictionary of already constructed nodes until we have built the node. This fact means that we cannot use the updated dictionary when building the descendants of the node -- which need the updated dictionary to link back. The problem can be overcome however with a ''credit card transform'' (a.k.a. "buy now, pay later" transform). To avoid hitting the bottom, we just have to "pay" by the "due date".<br />
<br />
For illustration, we will consider the problem of printing out a non-deterministic finite automaton (NFA) and transforming it into a deterministic finite automaton (DFA). Both NFA and DFA are represented as cyclic graphs. The problem has been discussed on the Haskell/Haskell-Cafe mailing lists. The automata in question were to recognize strings over a binary alphabet.<br />
<br />
A state of an automaton over a binary alphabet is a data structure:<br />
<haskell><br />
data (Ord l,Show l) => FaState l =<br />
FaState {label :: l, acceptQ :: Bool,<br />
trans0:: [FaState l],<br />
trans1:: [FaState l]}<br />
</haskell><br />
whose fields have the obvious meaning. Label is used for printing out and comparing states. The flag acceptQ tells if the state is final. Since an FaState can generally represent a non-deterministic automaton, transitions are the ''lists'' of states.<br />
<br />
An automaton is then a list of starting states.<br />
<haskell><br />
type FinAu l = [FaState l]<br />
</haskell><br />
For example, an automaton equivalent to the regular expression "0*(0(0+1)*)*" could be defined as:<br />
<haskell><br />
dom18 = [one]<br />
where one = FaState 1 True [one,two] []<br />
two = FaState 2 True [two,one] [one,two]<br />
</haskell><br />
using the straightforward translation from a regular expression to an NFA.<br />
<br />
We would like to compare and print automata and their states:<br />
<haskell><br />
instance (Ord l,Show l) => Eq (FaState l) where<br />
(FaState l1 _ _ _) == (FaState l2 _ _ _) = l1 == l2<br />
</haskell><br />
<br />
Printing a FaState however poses a slight problem. For example, the state labeled '1' in the automaton dom18 refers to itself. If we blindly 'follow the links', we will loop forever. Therefore, we must keep track of the already printed states. We need a data structure for such an occurrence check, with the following obvious operations:<br />
<br />
<haskell><br />
class OCC occ where<br />
empty:: occ a<br />
seenp:: (Eq a) => a -> occ a -> Bool -- occurrence check predicate<br />
put:: a -> occ a -> occ a -- add an item<br />
</haskell><br />
<br />
In this article, we realize such a data structure as a list. In the future, we can pull in something fancier from the Edison collection:<br />
<br />
<haskell><br />
instance OCC [] where<br />
empty = []<br />
seenp = elem<br />
put = (:)<br />
</haskell><br />
<br />
We are now ready to print an automaton. To be more precise, we traverse the corresponding graph depth-first, pre-order, and keep track of the already printed states. A 'states_seen' datum accumulates the shown states, so we can be sure we print each state only once and thus avoid the looping.<br />
<br />
<haskell><br />
instance (Ord l,Show l) => Show (FaState l) where<br />
show state = "{@" ++ showstates [state] (empty::[FaState l]) "@}"<br />
where<br />
-- showstates worklist seen_states suffix<br />
showstates [] states_seen suffix = suffix<br />
showstates (st:rest) states_seen suffix<br />
| st `seenp` states_seen = showstates rest states_seen suffix<br />
showstates (st@(FaState l accept t0 t1):rest) states_seen suffix =<br />
showstate st<br />
$ showstates (t0++t1++rest) (st `put` states_seen) suffix<br />
<br />
showstate (FaState l accept t0 t1) suffix<br />
= "{State " ++ (show l) ++<br />
" " ++ (show accept) ++ " " ++ (show $ map label t0) ++<br />
" " ++ (show $ map label t1) ++ "}" ++ suffix<br />
</haskell><br />
<br />
Now,<br />
<br />
<haskell><br />
CCardFA> print dom18 -- prints as<br />
CCardFA> [{@{State 1 True [1,2] []}{State 2 True [2,1] [1,2]}@}]<br />
</haskell><br />
<br />
The acceptance function for our automata can be written as follows. The function takes the list of starting states and the string over the boolean alphabet. The function returns True if the string is accepted.<br />
<br />
<haskell><br />
finAuAcceptStringQ start_states str =<br />
any (\l -> acceptP l str) start_states<br />
where acceptP (FaState _ acceptQ _ _) [] = acceptQ<br />
acceptP (FaState _ _ t0 t1) (s:rest) =<br />
finAuAcceptStringQ (if s then t1 else t0) rest<br />
</haskell><br />
<br />
To test the automata, we can try<br />
<br />
<haskell><br />
test1= finAuAcceptStringQ dom18 $ map (>0) [0,1,0,1]<br />
test2= finAuAcceptStringQ dom18 $ map (>0) [1,1,0,1]<br />
test3= finAuAcceptStringQ dom18 [True]<br />
test4= finAuAcceptStringQ dom18 [False]<br />
</haskell><br />
<br />
We are now ready to write the NFA->DFA conversion, a determinization of an NFA. We implement the textbook algorithm of tracing set of NFA states. A state in the resulting DFA corresponds to a list of the NFA states. A DFA is generally a cyclic graph, often with cycles of length 1 (self-referenced nodes). To be able to "link back" as we build DFA states, we have to remember the already constructed states. We need a data structure, a dictionary of states:<br />
<br />
<haskell><br />
class StateDict sd where<br />
emptyd :: sd (l,FaState l)<br />
locate :: (Eq l) => l -> sd (l,FaState l) -> Maybe (FaState l)<br />
putd :: (l,FaState l) -> sd (l,FaState l) -> sd (l,FaState l)<br />
</haskell><br />
<br />
For now, we realize this dictionary as an associative list. If performance matters, we can use a fancier dictionary from the Edison<br />
<br />
<haskell><br />
instance StateDict [] where<br />
emptyd = []<br />
locate = lookup<br />
putd = (:)<br />
</haskell><br />
<br />
The work of the NFA->DFA conversion is done by the following function determinize_cc. The function takes a list of NFA states, the dictionary of the already built states, and returns a pair <hask>([dfa_state], updated_dictionary)</hask> where <hask>[dfa_state]</hask> is a singleton list.<br />
<br />
<haskell><br />
-- [nfa_state] -> dictionary_of_seen_states -><br />
-- ([dfa_state],updated_dictionary)<br />
-- [dfa_state] is a singleton list<br />
determinize_cc states converted_states =<br />
-- first, check the cache to see if the state has been built already<br />
case dfa_label `locate` converted_states of<br />
Nothing -> build_state<br />
Just dfa_state -> ([dfa_state],converted_states)<br />
where<br />
-- [NFA_labels] -> DFA_labels<br />
det_labels = sort . nub . map label<br />
dfa_label = det_labels states<br />
<br />
-- find out NFA-followers for [nfa_state] upon ingestion of 0 and 1<br />
(t0_followers,t1_followers) =<br />
foldr (\st (f0,f1) -> (trans0 st ++ f0, trans1 st ++ f1))<br />
([],[]) states<br />
acceptQ' = any acceptQ states<br />
<br />
-- really build the dfa state and return ([dfa_state],updated_cache)<br />
build_state = let<br />
-- note, the dfa_state is computed _below_<br />
converted_states1 = (dfa_label,dfa_state) `putd` converted_states<br />
(t0', converted_states2) =<br />
(determinize_cc t0_followers converted_states1)<br />
(t1', converted_states3) =<br />
(determinize_cc t1_followers converted_states2)<br />
dfa_state =<br />
(FaState dfa_label acceptQ' t0' t1')<br />
in ([dfa_state],converted_states3)<br />
</haskell><br />
The front end of the NFA->DFA transformer:<br />
<hask>finAuDeterminize states = fst $ determinize_cc states []</hask><br />
<br />
At the heart of the credit card transform is the phrase from the above code:<br />
<hask> converted_states1 = (dfa_label,dfa_state) `putd` converted_states</hask><br />
<br />
The phrase expresses the addition to the dictionary of the 'converted_states' of a 'dfa_state' that we haven't built yet. The computation of the 'dfa_state' is written 4 lines below the phrase in question. Because (,) is non-strict in its arguments and locate is non-strict in its result, we can get away with a mere promise to "pay". Note that the computation of the dfa_state needs t0' and t1', which in turn rely on 'converted_states1'. This fact shows that we can tie the knot by making a promise to compute a state, add this promise to the dictionary of the built states, and use the updated dictionary to build the descendants. Because Haskell is a non-strict language, we don't need to do anything special to make the promise. Every computation is Haskell is by default a promise.<br />
<br />
We can print the DFA for dom18 to see what we've got:<br />
<haskell><br />
CCardFA> finAuDeterminize dom18<br />
CCardFA>-- which shows<br />
CCardFA> [{@{State [1] True [[1,2]] [[]] }<br />
CCardFA> {State [1,2] True [[1,2]] [[1,2]]}<br />
CCardFA> {State [] False [[]] [[]] }@}]<br />
</haskell><br />
which is indeed a DFA (which happens to be minimal) recognizing (0+1)* - 1(0+1)*<br />
<br />
We can run the determinized FA using the same function finAuAcceptStringQ:<br />
<haskell><br />
test1' = finAuAcceptStringQ (finAuDeterminize dom18) $ map (>0) [0,1,0,1]<br />
test2' = finAuAcceptStringQ (finAuDeterminize dom18) $ map (>0) [1,1,0,1]<br />
</haskell><br />
<br />
The complete code for this example is in http://pobox.com/~oleg/ftp/Haskell/CCard-transform-DFA.lhs. Another example of tying a knot in the case of forward links, by using a fixed-point combinator, is discussed in http://www.mail-archive.com/haskell@haskell.org/msg10687.html.<br />
<br />
-- Oleg<br />
<br />
For a long time, I've had an issue with Oleg's reply to Hal Daume III, the "forward links" example. The problem is that it doesn't really exploit laziness or circular values. It's solution would work even in a strict language. It's simply a functional version of the standard approach: build the result with markers and patch it up afterwards. It is a fairly clever way of doing purely something that is typically done with references and mutable update, but it doesn't really address what Hal Daume III was after. Fixing Hal Daume's example so that it won't loop is relatively trivial; simply change the case to a let or equivalently use a lazy pattern match in the case. However, if that's all there was to it, I would've written this a long time ago.<br />
<br />
The problem is that it no longer gives you control of the error message or anyway to recover from it. With GHC's extensions to exception handling you could do it, but you'd have to put readDecisionTree in the IO monad to recover from it, and if you wanted better messages you'd have to put most of the parsing in the IO monad so that you could catch the error earlier and provide more information then rethrow. What's kept me is that I couldn't figure out a way to tie the knot when the environment had a type like, Either String [(String,DecisionTree)]. This is because it's impossible for this case; we decide whether to return Left "could not find subtree" or Right someValue and therefore whether the environment is Left or Right based on whether we could find the subtree in the environment. In effect, we need to lookup a value in an environment we may return to know whether to return it. Obviously this is a truly circular dependency. This made me think that Oleg's solution was as good as any other and better than some (actually, ironically Oleg's solution also uses a let instead of a case, however, there's nothing stopping it from being a case, but it still would provide no way to recover from it without effectively doing what is mentioned below). Recently, I've thought about this again and the solution is obvious and follows directly from the original definition modified to use let. It doesn't loop because only particular values in the lookup table fail, in fact, you might never know there was a variable lookup error if you didn't touch all of the tree. This translates directly into the environment having type [(String,Either String DecisionTree)]. There are several benefits to this approach compared to Oleg's: 1) it solves my original problem, you are now able to specify the error messages (Oleg's can do this), 2) it goes beyond that (and beyond Hal Daume's original "specification") and also allows you to recover from an error without resorting to the IO monad and/or extensions (Oleg's can't do this), 3) it does implicitly what Oleg's version does explicitly, 4) because of (3) it shares properly while Oleg's does not*, 5) both the environment and the returned value are made up of showable values, not opaque functions, 6) it requires less changes to the original code and is more localized than Oleg's solution; only the variable lookup and top-level function will need to change.<br />
<br />
To recover, all one needs to do is make sure all the values in the lookup table are Right values. If they aren't, there are various ways you could collect the information; there are also variations on how to combine error information and what to provide. Even without a correctness check, you can still provide better error messages for the erroneous thunks. A possible variation that loses some of the benefits, is to change the DecisionTree type (or have a different version, [[IndirectComposite]] comes to mind here) that has Either ErrorInfo ErrorDecisionTree subnodes, which will allow you to recover at any time (though, if you want to make a normal DecisionTree out of it you will lose sharing). Also, the circular dependency only comes up if you need to use the environment to decide on an error. For example, a plain old syntactic parse error can cyclicly use an Either ErrorInfo [(String,DecisionTree)] perfectly fine (pass in fromRight env where fromRight ~(Right x) = x). It will also work even with the above approach giving the environment the type Either [(String,Either ErrorInfo DecisionTree)]. Below is code for a simplified scenario that does most of these things,<br />
<br />
<haskell><br />
module Main where<br />
import Maybe ( fromJust )<br />
import Monad<br />
<br />
main :: IO ()<br />
main = do<br />
input <- getContents<br />
length input `seq` print (fixup input)<br />
<br />
instance Monad (Either s) where<br />
return = Right<br />
m >>= f = either Left f m<br />
<br />
isLeft :: Either a b -> Bool<br />
isLeft (Left _) = True<br />
isLeft _ = False<br />
<br />
fromRight :: Either a b -> b<br />
fromRight ~(Right x) = x<br />
<br />
fixup :: String -> Either String [(String,Either String String)]<br />
fixup input = env<br />
where env = mapM (parse (fromRight env) . words) (lines input)<br />
<br />
checkedFixup :: String -> Either String [(String,String)]<br />
checkedFixup input =<br />
case fixup input of<br />
Left err -> Left err<br />
Right env -> case filter (isLeft . snd) env of<br />
[] -> Right $ map (\(n,Right v) -> (n,v)) env<br />
(_,Left err):_ -> Left err<br />
<br />
parse :: [(String,Either String String)] -> [String] -> Either String (String,Either String String)<br />
parse env ("define":name:values) = Right (name,values')<br />
where values' = liftM unwords $ mapM lookupRef values<br />
lookupRef ('*':word)<br />
= maybe (Left $ "couldn't find "++word++" in "++name)<br />
id<br />
(lookup word env)<br />
lookupRef word = Right word<br />
parse env input = Left $ "parse error with: "++unwords input<br />
</haskell><br />
checkedFixup demonstrates how you could check and recover, but since the environment is the return value neither fixup or checkedFixup quite illustrate having potentially erroneous thunks in the actual return value. Some example input,<br />
<haskell><br />
define x *y *y<br />
define y a b<br />
outputs Right [("x",Right "a b a b"),("y",Right "a b")]<br />
define x *y *y<br />
aousht<br />
define y a b<br />
outputs Left "parse error with: aousht"<br />
define x *y *z<br />
define y a b<br />
define z *w<br />
outputs Right [("x",Left "couldn't find w in z"),("y",Right "a b"),("z",Left "couldn't find w in z")]<br />
</haskell><br />
<br />
* Consider a tree Y that contains the subtree X twice. With Oleg's version, when we resolve the "X" variable we look up a (manually) delayed tree and then build X. Each subtree of Y will build it's own version of X. With the truly circular version each subtree of Y will be the same, possibly erroneous, thunk that builds X, if the thunk isn't erroneous then when it is updated both of Y's subtrees will point to the same X.<br />
<br />
-- DerekElkins<br />
<br />
<br />
[[Category:Code]]<br />
[[Category:Idioms]]</div>Blackouthttps://wiki.haskell.org/index.php?title=GHC/Typed_holes&diff=55364GHC/Typed holes2013-02-01T08:50:16Z<p>Blackout: Markup, removed asterisks not referencing anything</p>
<hr />
<div>[[Category:GHC|Type system]]<br />
<span style='font-size: x-large; font-weight: bold'>Type holes in GHC.</span><br />
<br />
Type holes are a powerful feature in GHC inspired by [http://wiki.portal.chalmers.se/agda/pmwiki.php Agda]. But what are type holes, and how do they help us write code?<br />
<br />
(This page was taken from a 'tutorial' I wrote of this feature [http://www.reddit.com/r/haskell/comments/10u7xr/ghc_head_now_features_agdalike_holes/c6gr9qe on Reddit].)<br />
<br />
== Introduction ==<br />
<br />
Sometimes, when writing programs, we tend to use lots of polymorphic types. That's good! We can reuse those pieces, thanks to the types. But it can be difficult sometimes, when writing a program, to see how the polymorphic types should exactly fit together, given the things in scope.<br />
<br />
Would it not be nice if we could have the compiler tell us the types of everything in scope? It'd be much easier to see how we can 'fit' them together like puzzle pieces.<br />
<br />
This is the purpose of a hole: it has similar semantics to <tt>undefined</tt>, in that evaluating it is an error (you can replace all holes with <tt>undefined</tt>, and vice versa, and nothing has changed: you may even say <tt>undefined</tt> is just a really crappy, useless hole!) But it's special in that, when GHC encounters a hole during compilation, it will tell you what type needs to be there in place of the hole, for the type-checker to be OK with the definition. It will also show you the types of all the bindings in scope, to assist you in figuring out the problem.<br />
<br />
This feature was implemented by Simon Peyton Jones, Sean Leather and Thijs Alkemade, and is scheduled to be released with GHC 7.8.1 (likely released sometime mid-2013.) It is enabled with the <tt>TypeHoles</tt> language extension.<br />
<br />
== Motivating example ==<br />
<br />
Let us consider the definition of a Free monad:<br />
<br />
<haskell><br />
data Free f a<br />
= Pure a<br />
| Free (f (Free f a))<br />
</haskell><br />
<br />
A free monad coupled with an underlying <tt>Functor f</tt>. We would like to write the Monad instance for <tt>Free f</tt>.<br />
<br />
We begin by writing the trivial cases:<br />
<br />
<haskell><br />
instance Functor f => Monad (Free f) where<br />
return a = Pure a<br />
Pure a >>= f = f a<br />
Free f >>= g = ... -- I'm confused!<br />
</haskell><br />
<br />
... but we're a little stumped on how to write the final case. Let's have the compiler help us do this.<br />
<br />
== Writing 'instance Monad (Free f)' ==<br />
<br />
Let's go ahead and shove all of this in a file. For the final case of the <tt>Monad (Free f)</tt> instance, we will place a hole in the definition, which is designated by the same syntax for wildcard patterns, <tt>_</tt><br />
<br />
<haskell><br />
{-# LANGUAGE TypeHoles #-}<br />
module FreeMonad where<br />
<br />
data Free f a<br />
= Pure a<br />
| Free (f (Free f a))<br />
<br />
instance Functor f => Monad (Free f) where<br />
return a = Pure a<br />
Pure a >>= f = f a<br />
Free f >>= g = Free _<br />
</haskell><br />
<br />
Now let's load that into GHCi. We should see something that looks like this:<br />
<br />
<code><pre><br />
$ ghci FreeMonad.hs<br />
[1 of 1] Compiling Main ( holes.hs, interpreted )<br />
<br />
FreeMonad.hs:11:23: Warning:<br />
Found hole `_' with type f (Free f b)<br />
Where: `f' is a rigid type variable bound by<br />
the instance declaration at holes.hs:26:10<br />
`b' is a rigid type variable bound by<br />
the type signature for<br />
>>= :: Free f a -> (a -> Free f b) -> Free f b<br />
at FreeMonad.hs:10:10<br />
Relevant bindings include<br />
>>= :: Free f a -> (a -> Free f b) -> Free f b<br />
(bound at FreeMonad.hs:11:3)<br />
f :: f (Free f a) (bound at FreeMonad.hs:11:8)<br />
g :: a -> Free f b (bound at FreeMonad.hs:11:14)<br />
In the first argument of `Free', namely `_'<br />
In the expression: Free (_)<br />
In an equation for `>>=': (Free f) >>= g = Free (_)<br />
Ok, modules loaded: Main.<br />
*Main> <br />
</pre></code><br />
<br />
Let's look at the error, and what the hole should look like (first line of the error):<br />
<br />
<code><pre><br />
Found hole `_' with type f (Free f b)<br />
</pre></code><br />
<br />
OK, so I need some value of type <tt>f (Free f b)</tt> to put there. I don't know how to get that. But let's look at what's in scope:<br />
<br />
<code><pre><br />
Relevant bindings include<br />
>>= :: Free f a -> (a -> Free f b) -> Free f b<br />
(bound at holes.hs:28:3)<br />
f :: f (Free f a) (bound at holes.hs:29:8)<br />
g :: a -> Free f b (bound at holes.hs:29:14)<br />
In the first argument of `Free', namely `_'<br />
</pre></code><br />
<br />
OK, so I have 3 values directly surrounding me at my disposal: <tt>f</tt>, <tt>g</tt>, and <tt>(>>=)</tt>. I have a <tt>f :: f (Free f a)</tt> in scope, and that's almost what I need. I just need to turn the inner <tt>Free f a</tt> into a <tt>Free f b</tt>. And that's what fmap does for functors! And I don't know what the functor <tt>f</tt> is anyway, so clearly I need to <tt>fmap</tt> the value before anything else. Let's do that and use a hole to tell us where to go next:<br />
<br />
<code><pre><br />
instance Monad ... where<br />
...<br />
Free f >>= g = Free (fmap _ f)<br />
</pre></code><br />
<br />
Now we get from the compiler:<br />
<br />
<code><br />
Found hole `_' with type Free f a -> Free f b<br />
</code><br />
<br />
OK, so now we just need some function with type <tt>Free f a -> Free f b</tt>. Let's look at what's in scope. We can quickly see that <tt>(>>=)</tt> has the type we want if we partially apply it. And <tt>g</tt> has just the type we need for that partial application! Equationally:<br />
<br />
<code><pre><br />
(>>=) :: Free f a -> (a -> Free f b) -> Free f b<br />
g :: a -> Free f b<br />
</pre></code><br />
<br />
Ergo:<br />
<br />
<code><pre><br />
(\x -> x >>= g) :: Free f a -> Free f b<br />
</pre></code><br />
<br />
So the resulting instance is:<br />
<br />
<code><pre><br />
instance Functor f => Monad (Free f) where<br />
return a = Pure a<br />
Pure a >>= f = f a<br />
Free f >>= g = Free (fmap (>>= g) f)<br />
</pre></code><br />
<br />
== Notes on parametricity, free theorems and Agda ==<br />
<br />
These examples hint at what people mean when they say they use the type system to drive development: it can help write your code, too!<br />
<br />
This is of course very powerful for general, polymorphic types like above, because a polymorphic function may only be written so many ways. In fact, the above definition is the only legitimate <tt>Monad</tt> instance for <tt>Free</tt>!<br />
<br />
A polymorphic type variable says what you cannot do to it: touch! You can't break the parametric nature of it by scrutinizing the variable in any way. A useful implementation of a polymorphic function has a small 'design space', naturally. It is not so, when defining monomorphic functions. A function like <tt>f :: Int -> Double</tt> could have many possible definitions, but we can always be sure for example, that <tt>f :: a -> a</tt> only has one valid definition, or that if <tt>f :: [a] -> [a]</tt>, then 'trivially' we can see that <tt>f [] = []</tt> must hold as a law.<br />
<br />
This notion of "what information may be calculated based on the type" like we're doing here can be formalized to what we call 'Free Theorems', pioneered by Philip Wadler in his paper '[http://ttic.uchicago.edu/~dreyer/course/papers/wadler.pdf Theorems for Free!]'<br />
<br />
Free theorems fall out of the parametricity theorem (also called Reynolds' abstraction theorem,) which essentially states how polymorphic types relate to each other. Free theorems just state that, given a polymorphic type, it's possible to derive useful theorems and laws, based only on that type alone (so if <tt>f :: [a] -> [a]</tt> as we said earlier, then we get the free theorem <tt>f [] = []</tt> by definition - try writing that case another way! You can't!)<br />
<br />
At this point, you're getting into much deeper territory, but I say all this because you have probably intuitively thought some of these things, when writing polymorphic functions yourself - that you must write them in such a way that the terms are all properly abstracted over correctly. This kind of design 'forces' only a few definitions to be possible.<br />
<br />
Holes for GHC were inspired by the equivalent feature in its (twice removed) sister language, [http://wiki.portal.chalmers.se/agda/pmwiki.php Agda]. Holes are considerably more powerful in Agda, mostly due to the fact it is total, the compiler infers less, and thus Agda programs are <i>rich</i> with evidence, giving witness to many facts the compiler can use. As a result, it can derive more code for you. In particular, Agda can literally <i>write</i> your code based on holes. It's able to do things like write case-analysis' for you, based only on type structure, and what's in scope, and discharge all the 'trivial' cases. You do this in emacs, in one or two keystrokes. It's essentially a semi-automated feedback loop for development, and it makes sense because the tool is really working with a meaningful model of your code, only in sensible ways that type-check. It's great.<br />
<br />
There is not necessarily a reason why this could not exist for e.g. the Emacs mode for Haskell. Just because Haskell is not total, doesn't mean we couldn't calculate useful code based on the type - it is just much harder, especially considering the multitudes of features and extensions GHC supports.<br />
<br />
* When I say 'legitimate' or 'useful', I mean, 'modulo divergent definitions.' Because Haskell is not total, it is possible to write divergent terms like <tt>let x = x in x</tt>, and the type of this expression is <tt>forall a. a</tt>, meaning I can prove <i>anything</i> with it. In the domain of Haskell semantics, this expression is in fact equivalent to <tt>undefined</tt> as well, because all bottoms are equal. I can prove that Blue is Green this way, or that Freedom is Slavery - I can inhabit any type, using these nonsensical terms. Logistically, you could say Haskell terms are complete, but not coherent/consistent. But divergent definitions here are not very legitimate, interesting, or useful, obviously, so ignore them (cue, [http://www.cse.chalmers.se/~nad/publications/danielsson-et-al-popl2006.html "Fast and Loose reasoning is morally correct."])</div>Blackouthttps://wiki.haskell.org/index.php?title=Typeclassopedia&diff=55115Typeclassopedia2012-12-21T17:59:36Z<p>Blackout: /* Comonad */ Updated Comonad to the newest library interface (version 3), i.e. no "copointed".</p>
<hr />
<div>''By [[User:Byorgey|Brent Yorgey]], byorgey@cis.upenn.edu''<br />
<br />
''Originally published 12 March 2009 in [http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf issue 13] of [http://themonadreader.wordpress.com/ the Monad.Reader]. Ported to the Haskell wiki in November 2011 by [[User:Geheimdienst|Geheimdienst]].''<br />
<br />
''This is now the official version of the Typeclassopedia and supersedes the version published in the Monad.Reader. Please help update and extend it by editing it yourself or by leaving comments, suggestions, and questions on the [[Talk:Typeclassopedia|talk page]].''<br />
<br />
=Abstract=<br />
<br />
The standard Haskell libraries feature a number of type classes with algebraic or category-theoretic underpinnings. Becoming a fluent Haskell hacker requires intimate familiarity with them all, yet acquiring this familiarity often involves combing through a mountain of tutorials, blog posts, mailing list archives, and IRC logs.<br />
<br />
The goal of this document is to serve as a starting point for the student of Haskell wishing to gain a firm grasp of its standard type classes. The essentials of each type class are introduced, with examples, commentary, and extensive references for further reading.<br />
<br />
=Introduction=<br />
<br />
Have you ever had any of the following thoughts?<br />
* What the heck is a monoid, and how is it different from a mon<u>a</u>d?<br />
<br />
* I finally figured out how to use [[Parsec]] with do-notation, and someone told me I should use something called <code>Applicative</code> instead. Um, what?<br />
<br />
* Someone in the [[IRC channel|#haskell]] IRC channel used <code>(***)</code>, and when I asked lambdabot to tell me its type, it printed out scary gobbledygook that didn’t even fit on one line! Then someone used <code>fmap fmap fmap</code> and my brain exploded.<br />
<br />
* When I asked how to do something I thought was really complicated, people started typing things like <code>zip.ap fmap.(id &&& wtf)</code> and the scary thing is that they worked! Anyway, I think those people must actually be robots because there’s no way anyone could come up with that in two seconds off the top of their head.<br />
<br />
If you have, look no further! You, too, can write and understand concise, elegant, idiomatic Haskell code with the best of them.<br />
<br />
There are two keys to an expert Haskell hacker’s wisdom:<br />
# Understand the types.<br />
# Gain a deep intuition for each type class and its relationship to other type classes, backed up by familiarity with many examples.<br />
<br />
It’s impossible to overstate the importance of the first; the patient student of type signatures will uncover many profound secrets. Conversely, anyone ignorant of the types in their code is doomed to eternal uncertainty. “Hmm, it doesn’t compile ... maybe I’ll stick in an<br />
<code>fmap</code> here ... nope, let’s see ... maybe I need another <code>(.)</code> somewhere? ... um ...”<br />
<br />
The second key—gaining deep intuition, backed by examples—is also important, but much more difficult to attain. A primary goal of this document is to set you on the road to gaining such intuition. However—<br />
<br />
:''There is no royal road to Haskell. {{h:title|Well, he probably would have said it if he knew Haskell.|—Euclid}}''<br />
<br />
This document can only be a starting point, since good intuition comes from hard work, [http://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/ not from learning the right metaphor]. Anyone who reads and understands all of it will still have an arduous journey ahead—but sometimes a good starting point makes a big difference.<br />
<br />
It should be noted that this is not a Haskell tutorial; it is assumed that the reader is already familiar with the basics of Haskell, including the standard <code>[http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html Prelude]</code>, the type system, data types, and type classes.<br />
<br />
The type classes we will be discussing and their interrelationships:<br />
<br />
[[Image:Typeclassopedia-diagram.png]]<br />
<br />
{{note|<code>Semigroup</code> can be found in the [http://hackage.haskell.org/package/semigroups <code>semigroups</code> package], <code>Apply</code> in the [http://hackage.haskell.org/package/semigroupoids <code>semigroupoids</code> package], and <code>Comonad</code> in the [http://hackage.haskell.org/package/comonad <code>comonad</code> package].}}<br />
<br />
* <span style="border-bottom: 2px solid black">Solid arrows</span> point from the general to the specific; that is, if there is an arrow from <code>Foo</code> to <code>Bar</code> it means that every <code>Bar</code> is (or should be, or can be made into) a <code>Foo</code>.<br />
* <span style="border-bottom: 2px dotted black">Dotted arrows</span> indicate some other sort of relationship.<br />
* <code>Monad</code> and <code>ArrowApply</code> are equivalent.<br />
* <code>Semigroup</code>, <code>Apply</code> and <code>Comonad</code> are greyed out since they are not actually (yet?) in the standard Haskell libraries {{noteref}}.<br />
<br />
One more note before we begin. The original spelling of “type class” is with two words, as evidenced by, for example, the [http://haskell.org/onlinereport/ Haskell 98 Revised Report], early papers on type classes like [http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.103.5639 Type classes in Haskell] and [http://research.microsoft.com/en-us/um/people/simonpj/papers/type-class-design-space/ Type classes: exploring the design space], and [http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.168.4008 Hudak et al.’s history of Haskell]. However, as often happens with two-word phrases that see a lot of use, it has started to show up as one word (“typeclass”) or, rarely, hyphenated (“type-class”). When wearing my prescriptivist hat, I prefer “type class”, but realize (after changing into my descriptivist hat) that there's probably not much I can do about it.<br />
<br />
We now begin with the simplest type class of all: <code>Functor</code>.<br />
<br />
=Functor=<br />
<br />
The <code>Functor</code> class ([http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#t:Functor haddock]) is the most basic and ubiquitous type class in the Haskell libraries. A simple intuition is that a <code>Functor</code> represents a “container” of some sort, along with the ability to apply a function uniformly to every element in the container. For example, a list is a container of elements, and we can apply a function to every element of a list, using <code>map</code>. As another example, a binary tree is also a container of elements, and it’s not hard to come up with a way to recursively apply a function to every element in a tree.<br />
<br />
Another intuition is that a <code>Functor</code> represents some sort of “computational context”. This intuition is generally more useful, but is more difficult to explain, precisely because it is so general. Some examples later should help to clarify the <code>Functor</code>-as-context point of view.<br />
<br />
In the end, however, a <code>Functor</code> is simply what it is defined to be; doubtless there are many examples of <code>Functor</code> instances that don’t exactly fit either of the above intuitions. The wise student will focus their attention on definitions and examples, without leaning too heavily on any particular metaphor. Intuition will come, in time, on its own.<br />
<br />
==Definition==<br />
<br />
Here is the type class declaration for <code>Functor</code>:<br />
<br />
<haskell><br />
class Functor f where<br />
fmap :: (a -> b) -> f a -> f b<br />
</haskell><br />
<br />
<code>Functor</code> is exported by the <code>Prelude</code>, so no special imports are needed to use it.<br />
<br />
First, the <code>f a</code> and <code>f b</code> in the type signature for <code>fmap</code> tell us that <code>f</code> isn’t just a type; it is a ''type constructor'' which takes another type as a parameter. (A more precise way to say this is that the ''kind'' of <code>f</code> must be <code>* -> *</code>.) For example, <code>Maybe</code> is such a type constructor: <code>Maybe</code> is not a type in and of itself, but requires another type as a parameter, like <code>Maybe Integer</code>. So it would not make sense to say <code>instance Functor Integer</code>, but it could make sense to say <code>instance Functor Maybe</code>.<br />
<br />
Now look at the type of <code>fmap</code>: it takes any function from <code>a</code> to <code>b</code>, and a value of type <code>f a</code>, and outputs a value of type <code>f b</code>. From the container point of view, the intention is that <code>fmap</code> applies a function to each element of a container, without altering the structure of the container. From the context point of view, the intention is that <code>fmap</code> applies a function to a value without altering its context. Let’s look at a few specific examples.<br />
<br />
==Instances==<br />
<br />
{{note|Recall that <code>[]</code> has two meanings in Haskell: it can either stand for the empty list, or, as here, it can represent the list type constructor (pronounced “list-of”). In other words, the type <code>[a]</code> (list-of-<code>a</code>) can also be written <code>[] a</code>.}}<br />
<br />
{{note|You might ask why we need a separate <code>map</code> function. Why not just do away with the current list-only <code>map</code> function, and rename <code>fmap</code> to <code>map</code> instead? Well, that’s a good question. The usual argument is that someone just learning Haskell, when using <code>map</code> incorrectly, would much rather see an error about lists than about <code>Functor</code>s.}}<br />
<br />
As noted before, the list constructor <code>[]</code> is a functor {{noteref}}; we can use the standard list function <code>map</code> to apply a function to each element of a list {{noteref}}. The <code>Maybe</code> type constructor is also a functor, representing a container which might hold a single element. The function <code>fmap g</code> has no effect on <code>Nothing</code> (there are no elements to which <code>g</code> can be applied), and simply applies <code>g</code> to the single element inside a <code>Just</code>. Alternatively, under the context interpretation, the list functor represents a context of nondeterministic choice; that is, a list can be thought of as representing a single value which is nondeterministically chosen from among several possibilities (the elements of the list). Likewise, the <code>Maybe</code> functor represents a context with possible failure. These instances are:<br />
<br />
<haskell><br />
instance Functor [] where<br />
fmap _ [] = []<br />
fmap g (x:xs) = g x : fmap g xs<br />
-- or we could just say fmap = map<br />
<br />
instance Functor Maybe where<br />
fmap _ Nothing = Nothing<br />
fmap g (Just a) = Just (g a)<br />
</haskell><br />
<br />
As an aside, in idiomatic Haskell code you will often see the letter <code>f</code> used to stand for both an arbitrary <code>Functor</code> and an arbitrary function. In this document, <code>f</code> represents only <code>Functor</code>s, and <code>g</code> or <code>h</code> always represent functions, but you should be aware of the potential confusion. In practice, what <code>f</code> stands for should always be clear from the context, by noting whether it is part of a type or part of the code.<br />
<br />
There are other <code>Functor</code> instances in the standard libraries; below are a few. Note that some of these instances are not exported by the <code>Prelude</code>; to access them, you can import <code>Control.Monad.Instances</code>.<br />
<br />
* <code>Either e</code> is an instance of <code>Functor</code>; <code>Either e a</code> represents a container which can contain either a value of type <code>a</code>, or a value of type <code>e</code> (often representing some sort of error condition). It is similar to <code>Maybe</code> in that it represents possible failure, but it can carry some extra information about the failure as well.<br />
<br />
* <code>((,) e)</code> represents a container which holds an “annotation” of type <code>e</code> along with the actual value it holds. It might be clearer to write it as <code>(e,)</code>, by analogy with an operator section like <code>(1+)</code>, but that syntax is not allowed in types (although it is allowed in expressions with the <code>TupleSections</code> extension enabled). However, you can certainly ''think'' of it as <code>(e,)</code>.<br />
<br />
* <code>((->) e)</code> (which can be thought of as <code>(e ->)</code>; see above), the type of functions which take a value of type <code>e</code> as a parameter, is a <code>Functor</code>. As a container, <code>(e -> a)</code> represents a (possibly infinite) set of values of <code>a</code>, indexed by values of <code>e</code>. Alternatively, and more usefully, <code>((->) e)</code> can be thought of as a context in which a value of type <code>e</code> is available to be consulted in a read-only fashion. This is also why <code>((->) e)</code> is sometimes referred to as the ''reader monad''; more on this later.<br />
<br />
* <code>IO</code> is a <code>Functor</code>; a value of type <code>IO a</code> represents a computation producing a value of type <code>a</code> which may have I/O effects. If <code>m</code> computes the value <code>x</code> while producing some I/O effects, then <code>fmap g m</code> will compute the value <code>g x</code> while producing the same I/O effects.<br />
<br />
* Many standard types from the [http://hackage.haskell.org/package/containers/ containers library] (such as <code>Tree</code>, <code>Map</code>, and <code>Sequence</code>) are instances of <code>Functor</code>. A notable exception is <code>Set</code>, which cannot be made a <code>Functor</code> in Haskell (although it is certainly a mathematical functor) since it requires an <code>Ord</code> constraint on its elements; <code>fmap</code> must be applicable to ''any'' types <code>a</code> and <code>b</code>. However, <code>Set</code> (and other similarly restricted data types) can be made an instance of a suitable generalization of <code>Functor</code>, either by [http://article.gmane.org/gmane.comp.lang.haskell.cafe/78052/ making <code>a</code> and <code>b</code> arguments to the <code>Functor</code> type class themselves], or by adding an [http://blog.omega-prime.co.uk/?p=127 associated constraint].<br />
<br />
{{Exercises|<br />
<ol><br />
<li>Implement <code>Functor</code> instances for <code>Either e</code> and <code>((->) e)</code>.</li><br />
<li>Implement <code>Functor</code> instances for <code>((,) e)</code> and for <code>Pair</code>, defined as <br />
<br />
<haskell>data Pair a = Pair a a</haskell><br />
<br />
Explain their similarities and differences.<br />
</li><br />
<li>Implement a <code>Functor</code> instance for the type <code>ITree</code>, defined as<br />
<br />
<haskell><br />
data ITree a = Leaf (Int -> a) <br />
| Node [ITree a]<br />
</haskell><br />
</li><br />
<li>Give an example of a type of kind <code>* -> *</code> which cannot be made an instance of <code>Functor</code> (without using <code>undefined</code>).<br />
</li><br />
<li>Is this statement true or false? <br />
<br />
:''The composition of two <code>Functor</code>s is also a <code>Functor</code>.''<br />
<br />
If false, give a counterexample; if true, prove it by exhibiting some appropriate Haskell code.<br />
</li><br />
</ol><br />
}}<br />
<br />
==Laws==<br />
<br />
As far as the Haskell language itself is concerned, the only requirement to be a <code>Functor</code> is an implementation of <code>fmap</code> with the proper type. Any sensible <code>Functor</code> instance, however, will also satisfy the ''functor laws'', which are part of the definition of a mathematical functor. There are two:<br />
<br />
<haskell><br />
fmap id = id<br />
fmap (g . h) = (fmap g) . (fmap h)<br />
</haskell><br />
<br />
{{note|Technically, these laws make <code>f</code> and <code>fmap</code> together an endofunctor on ''Hask'', the category of Haskell types (ignoring [[Bottom|&perp;]], which is a party pooper). See [http://en.wikibooks.org/wiki/Haskell/Category_theory Wikibook: Category theory].}}<br />
<br />
Together, these laws ensure that <code>fmap g</code> does not change the ''structure'' of a container, only the elements. Equivalently, and more simply, they ensure that <code>fmap g</code> changes a value without altering its context {{noteref}}.<br />
<br />
The first law says that mapping the identity function over every item in a container has no effect. The second says that mapping a composition of two functions over every item in a container is the same as first mapping one function, and then mapping the other.<br />
<br />
As an example, the following code is a “valid” instance of <code>Functor</code> (it typechecks), but it violates the functor laws. Do you see why?<br />
<br />
<haskell><br />
-- Evil Functor instance<br />
instance Functor [] where<br />
fmap _ [] = []<br />
fmap g (x:xs) = g x : g x : fmap g xs<br />
</haskell><br />
<br />
Any Haskeller worth their salt would reject this code as a gruesome abomination.<br />
<br />
Unlike some other type classes we will encounter, a given type has at most one valid instance of <code>Functor</code>. This [http://article.gmane.org/gmane.comp.lang.haskell.libraries/15384 can be proven] via the [http://homepages.inf.ed.ac.uk/wadler/topics/parametricity.html#free ''free theorem''] for the type of <code>fmap</code>. In fact, [http://byorgey.wordpress.com/2010/03/03/deriving-pleasure-from-ghc-6-12-1/ GHC can automatically derive] <code>Functor</code> instances for many data types.<br />
<br />
A similar argument also shows that any <code>Functor</code> instance satisfying the first law (<code>fmap id = id</code>) will automatically satisfy the second law as well. Practically, this means that only the first law needs to be checked (usually by a very straightforward induction) to ensure that a <code>Functor</code> instance is valid.<br />
<br />
{{Exercises|<br />
# Although it is not possible for a <code>Functor</code> instance to satisfy the first <code>Functor</code> law but not the second, the reverse is possible. Give an example of a (bogus) <code>Functor</code> instance which satisfies the second law but not the first.<br />
# Which laws are violated by the evil <code>Functor</code> instance for list shown above: both laws, or the first law alone? Give specific counterexamples.<br />
}}<br />
<br />
==Intuition==<br />
<br />
There are two fundamental ways to think about <code>fmap</code>. The first has already been mentioned: it takes two parameters, a function and a container, and applies the function “inside” the container, producing a new container. Alternately, we can think of <code>fmap</code> as applying a function to a value in a context (without altering the context).<br />
<br />
Just like all other Haskell functions of “more than one parameter”, however, <code>fmap</code> is actually ''curried'': it does not really take two parameters, but takes a single parameter and returns a function. For emphasis, we can write <code>fmap</code>’s type with extra parentheses: <code>fmap :: (a -> b) -> (f a -> f b)</code>. Written in this form, it is apparent that <code>fmap</code> transforms a “normal” function (<code>g :: a -> b</code>) into one which operates over containers/contexts (<code>fmap g :: f a -> f b</code>). This transformation is often referred to as a ''lift''; <code>fmap</code> “lifts” a function from the “normal world” into the “<code>f</code> world”.<br />
<br />
==Further reading==<br />
<br />
A good starting point for reading about the category theory behind the concept of a functor is the excellent [http://en.wikibooks.org/wiki/Haskell/Category_theory Haskell wikibook page on category theory].<br />
<br />
=Applicative=<br />
<br />
A somewhat newer addition to the pantheon of standard Haskell type classes, ''applicative functors'' represent an abstraction lying in between <code>Functor</code> and <code>Monad</code> in expressivity, first described by McBride and Paterson. The title of their classic paper, [http://www.soi.city.ac.uk/~ross/papers/Applicative.html Applicative Programming with Effects], gives a hint at the intended intuition behind the [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html <code>Applicative</code>] type class. It encapsulates certain sorts of “effectful” computations in a functionally pure way, and encourages an “applicative” programming style. Exactly what these things mean will be seen later.<br />
<br />
==Definition==<br />
<br />
Recall that <code>Functor</code> allows us to lift a “normal” function to a function on computational contexts. But <code>fmap</code> doesn’t allow us to apply a function which is itself in a context to a value in a context. <code>Applicative</code> gives us just such a tool, <code>(<*>)</code>. It also provides a method, <code>pure</code>, for embedding values in a default, “effect free” context. Here is the type class declaration for <code>Applicative</code>, as defined in <code>Control.Applicative</code>:<br />
<br />
<haskell><br />
class Functor f => Applicative f where<br />
pure :: a -> f a<br />
(<*>) :: f (a -> b) -> f a -> f b<br />
</haskell><br />
<br />
Note that every <code>Applicative</code> must also be a <code>Functor</code>. In fact, as we will see, <code>fmap</code> can be implemented using the <code>Applicative</code> methods, so every <code>Applicative</code> is a functor whether we like it or not; the <code>Functor</code> constraint forces us to be honest.<br />
<br />
{{note|Recall that <code>($)</code> is just function application: <code>f $ x {{=}} f x</code>.}}<br />
<br />
As always, it’s crucial to understand the type signatures. First, consider <code>(<*>)</code>: the best way of thinking about it comes from noting that the type of <code>(<*>)</code> is similar to the type of <code>($)</code> {{noteref}}, but with everything enclosed in an <code>f</code>. In other words, <code>(<*>)</code> is just function application within a computational context. The type of <code>(<*>)</code> is also very similar to the type of <code>fmap</code>; the only difference is that the first parameter is <code>f (a -> b)</code>, a function in a context, instead of a “normal” function <code>(a -> b)</code>.<br />
<br />
<code>pure</code> takes a value of any type <code>a</code>, and returns a context/container of type <code>f a</code>. The intention is that <code>pure</code> creates some sort of “default” container or “effect free” context. In fact, the behavior of <code>pure</code> is quite constrained by the laws it should satisfy in conjunction with <code>(<*>)</code>. Usually, for a given implementation of <code>(<*>)</code> there is only one possible implementation of <code>pure</code>.<br />
<br />
(Note that previous versions of the Typeclassopedia explained <code>pure</code> in terms of a type class <code>Pointed</code>, which can still be found in the [http://hackage.haskell.org/package/pointed <code>pointed</code> package]. However, the current consensus is that <code>Pointed</code> is not very useful after all. For a more detailed explanation, see [[Why not Pointed?]])<br />
<br />
==Laws==<br />
<br />
{{note|See<br />
[http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html haddock for Applicative] and [http://www.soi.city.ac.uk/~ross/papers/Applicative.html Applicative programming with effects]}}<br />
<br />
Traditionally, there are four laws that <code>Applicative</code> instances should satisfy {{noteref}}. In some sense, they are all concerned with making sure that <code>pure</code> deserves its name:<br />
<br />
* The identity law:<br /><haskell>pure id <*> v = v</haskell><br />
* Homomorphism:<br /><haskell>pure f <*> pure x = pure (f x)</haskell>Intuitively, applying a non-effectful function to a non-effectful argument in an effectful context is the same as just applying the function to the argument and then injecting the result into the context with <code>pure</code>.<br />
* Interchange:<br /><haskell>u <*> pure y = pure ($ y) <*> u</haskell>Intuitively, this says that when evaluating the application of an effectful function to a pure argument, the order in which we evaluate the function and its argument doesn't matter.<br />
* Composition:<br /><haskell>u <*> (v <*> w) = pure (.) <*> u <*> v <*> w </haskell>This one is the trickiest law to gain intuition for. In some sense it is expressing a sort of associativity property of <code>(<*>)</code>. The reader may wish to simply convince themselves that this law is type-correct.<br />
<br />
Considered as left-to-right rewrite rules, the homomorphism, interchange, and composition laws actually constitute an algorithm for transforming any expression using <code>pure</code> and <code>(<*>)</code> into a canonical form with only a single use of <code>pure</code> at the very beginning and only left-nested occurrences of <code>(<*>)</code>. Composition allows reassociating <code>(<*>)</code>; interchange allows moving occurrences of <code>pure</code> leftwards; and homomorphism allows collapsing multiple adjacent occurrences of <code>pure</code> into one.<br />
<br />
There is also a law specifying how <code>Applicative</code> should relate to <code>Functor</code>:<br />
<br />
<haskell><br />
fmap g x = pure g <*> x<br />
</haskell><br />
<br />
It says that mapping a pure function <code>g</code> over a context <code>x</code> is the same as first injecting <code>g</code> into a context with <code>pure</code>, and then applying it to <code>x</code> with <code>(<*>)</code>. In other words, we can decompose <code>fmap</code> into two more atomic operations: injection into a context, and application within a context. The <code>Control.Applicative</code> module also defines <code>(<$>)</code> as a synonym for <code>fmap</code>, so the above law can also be expressed as:<br />
<br />
<code>g <$> x = pure g <*> x</code>.<br />
<br />
{{Exercises|<br />
# (Tricky) One might imagine a variant of the interchange law that says something about applying a pure function to an effectful argument. Using the above laws, prove that<haskell>pure f <*> x = pure (flip ($)) <*> x <*> pure f</haskell><br />
}}<br />
<br />
==Instances==<br />
<br />
Most of the standard types which are instances of <code>Functor</code> are also instances of <code>Applicative</code>.<br />
<br />
<code>Maybe</code> can easily be made an instance of <code>Applicative</code>; writing such an instance is left as an exercise for the reader.<br />
<br />
The list type constructor <code>[]</code> can actually be made an instance of <code>Applicative</code> in two ways; essentially, it comes down to whether we want to think of lists as ordered collections of elements, or as contexts representing multiple results of a nondeterministic computation (see Wadler’s [http://www.springerlink.com/content/y7450255v2670167/ How to replace failure by a list of successes]).<br />
<br />
Let’s first consider the collection point of view. Since there can only be one instance of a given type class for any particular type, one or both of the list instances of <code>Applicative</code> need to be defined for a <code>newtype</code> wrapper; as it happens, the nondeterministic computation instance is the default, and the collection instance is defined in terms of a <code>newtype</code> called <code>ZipList</code>. This instance is:<br />
<br />
<haskell><br />
newtype ZipList a = ZipList { getZipList :: [a] }<br />
<br />
instance Applicative ZipList where<br />
pure = undefined -- exercise<br />
(ZipList gs) <*> (ZipList xs) = ZipList (zipWith ($) gs xs)<br />
</haskell><br />
<br />
To apply a list of functions to a list of inputs with <code>(<*>)</code>, we just match up the functions and inputs elementwise, and produce a list of the resulting outputs. In other words, we “zip” the lists together with function application, <code>($)</code>; hence the name <code>ZipList</code>. <br />
<br />
The other <code>Applicative</code> instance for lists, based on the nondeterministic computation point of view, is:<br />
<br />
<haskell><br />
instance Applicative [] where<br />
pure x = [x]<br />
gs <*> xs = [ g x | g <- gs, x <- xs ]<br />
</haskell><br />
<br />
Instead of applying functions to inputs pairwise, we apply each function to all the inputs in turn, and collect all the results in a list.<br />
<br />
Now we can write nondeterministic computations in a natural style. To add the numbers <code>3</code> and <code>4</code> deterministically, we can of course write <code>(+) 3 4</code>. But suppose instead of <code>3</code> we have a nondeterministic computation that might result in <code>2</code>, <code>3</code>, or <code>4</code>; then we can write<br />
<br />
<haskell><br />
pure (+) <*> [2,3,4] <*> pure 4<br />
</haskell><br />
<br />
or, more idiomatically,<br />
<br />
<haskell><br />
(+) <$> [2,3,4] <*> pure 4.<br />
</haskell><br />
<br />
There are several other <code>Applicative</code> instances as well:<br />
<br />
* <code>IO</code> is an instance of <code>Applicative</code>, and behaves exactly as you would think: to execute <code>m1 <*> m2</code>, first <code>m1</code> is executed, resulting in a function <code>f</code>, then <code>m2</code> is executed, resulting in a value <code>x</code>, and finally the value <code>f x</code> is returned as the result of executing <code>m1 <*> m2</code>.<br />
<br />
* <code>((,) a)</code> is an <code>Applicative</code>, as long as <code>a</code> is an instance of <code>Monoid</code> ([[#Monoid|section Monoid]]). The <code>a</code> values are accumulated in parallel with the computation.<br />
<br />
* The <code>Applicative</code> module defines the <code>Const</code> type constructor; a value of type <code>Const a b</code> simply contains an <code>a</code>. This is an instance of <code>Applicative</code> for any <code>Monoid a</code>; this instance becomes especially useful in conjunction with things like <code>Foldable</code> ([[#Foldable|section Foldable]]).<br />
<br />
* The <code>WrappedMonad</code> and <code>WrappedArrow</code> newtypes make any instances of <code>Monad</code> ([[#Monad|section Monad]]) or <code>Arrow</code> ([[#Arrow|section Arrow]]) respectively into instances of <code>Applicative</code>; as we will see when we study those type classes, both are strictly more expressive than <code>Applicative</code>, in the sense that the <code>Applicative</code> methods can be implemented in terms of their methods.<br />
<br />
{{Exercises|<br />
# Implement an instance of <code>Applicative</code> for <code>Maybe</code>.<br />
# Determine the correct definition of <code>pure</code> for the <code>ZipList</code> instance of <code>Applicative</code>—there is only one implementation that satisfies the law relating <code>pure</code> and <code>(<*>)</code>.<br />
}}<br />
<br />
==Intuition==<br />
<br />
McBride and Paterson’s paper introduces the notation <math>[[g \; x_1 \; x_2 \; \cdots \; x_n]]\ </math> to denote function application in a computational context. If each <math>x_i\ </math> has type <math>f \; t_i\ </math> for some applicative functor <math>f\ </math>, and <math>g\ </math> has type <math>t_1 \to t_2 \to \dots \to t_n \to t\ </math>, then the entire expression <math>[[g \; x_1 \; \cdots \; x_n]]\ </math> has type <math>f \; t\ </math>. You can think of this as applying a function to multiple “effectful” arguments. In this sense, the double bracket notation is a generalization of <code>fmap</code>, which allows us to apply a function to a single argument in a context.<br />
<br />
Why do we need <code>Applicative</code> to implement this generalization of <code>fmap</code>? Suppose we use <code>fmap</code> to apply <code>g</code> to the first parameter <code>x1</code>. Then we get something of type <code>f (t2 -> ... t)</code>, but now we are stuck: we can’t apply this function-in-a-context to the next argument with <code>fmap</code>. However, this is precisely what <code>(<*>)</code> allows us to do.<br />
<br />
This suggests the proper translation of the idealized notation <math>[[g \; x_1 \; x_2 \; \cdots \; x_n]]\ </math> into Haskell, namely<br />
<haskell><br />
g <$> x1 <*> x2 <*> ... <*> xn,<br />
</haskell><br />
<br />
recalling that <code>Control.Applicative</code> defines <code>(<$>)</code> as convenient infix shorthand for <code>fmap</code>. This is what is meant by an “applicative style”—effectful computations can still be described in terms of function application; the only difference is that we have to use the special operator <code>(<*>)</code> for application instead of simple juxtaposition.<br />
<br />
Note that <code>pure</code> allows embedding “non-effectful” arguments in the middle of an idiomatic application, like<br />
<haskell><br />
g <$> x1 <*> pure x2 <*> x3<br />
</haskell><br />
which has type <code>f d</code>, given<br />
<haskell><br />
g :: a -> b -> c -> d<br />
x1 :: f a<br />
x2 :: b<br />
x3 :: f c<br />
</haskell><br />
<br />
The double brackets are commonly known as “idiom brackets”, because they allow writing “idiomatic” function application, that is, function application that looks normal but has some special, non-standard meaning (determined by the particular instance of <code>Applicative</code> being used). Idiom brackets are not supported by GHC, but they are supported by the [http://personal.cis.strath.ac.uk/~conor/pub/she/ Strathclyde Haskell Enhancement], a preprocessor which (among many other things) translates idiom brackets into standard uses of <code>(<$>)</code> and <code>(<*>)</code>. This can result in much more readable code when making heavy use of <code>Applicative</code>.<br />
<br />
==Alternative formulation==<br />
<br />
An alternative, equivalent formulation of <code>Applicative</code> is given by<br />
<br />
<haskell><br />
class Functor f => Monoidal f where<br />
unit :: f ()<br />
(**) :: f a -> f b -> f (a,b)<br />
</haskell><br />
<br />
Intuitively, this states that a <i>monoidal</i> functor is one which has some sort of "default shape" and which supports some sort of "combining" operation. <code>pure</code> and <code>(<*>)</code> are equivalent in power to <code>unit</code> and <code>(**)</code> (see the Exercises below).<br />
<br />
Furthermore, to deserve the name "monoidal" (see the [[#Monoid|section on Monoids]]), instances of <code>Monoidal</code> ought to satisfy the following laws, which seem much more straightforward than the traditional <code>Applicative</code> laws:<br />
<br />
{{note|Here <code>f *** g {{=}} \x -> (f x, g x)</code>. See [[#Arrow|Arrows]].}}<br />
* Naturality{{noteref}}: <haskell>fmap (f *** g) (u ** v) = fmap f u ** fmap g v</haskell><br />
{{note|In this and the following laws, <code>≅</code> refers to <i>isomorphism</i> rather than equality. In particular we consider <code>(x,()) ≅ x ≅ ((),x)</code> and <code>((x,y),z) ≅ (x,(y,z))</code>.}}<br />
* Left identity{{noteref}}: <haskell>unit ** v ≅ v</haskell><br />
* Right identity: <haskell>u ** unit ≅ u</haskell><br />
* Associativity: <haskell>u ** (v ** w) ≅ (u ** v) ** w</haskell><br />
<br />
These turn out to be equivalent to the usual <code>Applicative</code> laws.<br />
<br />
Much of this section was taken from [http://blog.ezyang.com/2012/08/applicative-functors/ a blog post by Edward Z. Yang]; see his actual post for a bit more information.<br />
<br />
{{Exercises|<br />
# Implement <code>pure</code> and <code>(<*>)</code> in terms of <code>unit</code> and <code>(**)</code>, and vice versa.<br />
# (Tricky) Prove that given your implementations from the previous exercise, the usual <code>Applicative</code> laws and the <code>Monoidal</code> laws stated above are equivalent.<br />
}}<br />
<br />
==Further reading==<br />
<br />
There are many other useful combinators in the standard libraries implemented in terms of <code>pure</code> and <code>(<*>)</code>: for example, <code>(*>)</code>, <code>(<*)</code>, <code>(<**>)</code>, <code>(<$)</code>, and so on (see [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html haddock for Applicative]). Judicious use of such secondary combinators can often make code using <code>Applicative</code>s much easier to read.<br />
<br />
[http://www.soi.city.ac.uk/~ross/papers/Applicative.html McBride and Paterson’s original paper] is a treasure-trove of information and examples, as well as some perspectives on the connection between <code>Applicative</code> and category theory. Beginners will find it difficult to make it through the entire paper, but it is extremely well-motivated—even beginners will be able to glean something from reading as far as they are able.<br />
<br />
{{note|Introduced by [http://conal.net/papers/simply-reactive/ an earlier paper] that was since superseded by [http://conal.net/papers/push-pull-frp/ Push-pull functional reactive programming].}}<br />
<br />
Conal Elliott has been one of the biggest proponents of <code>Applicative</code>. For example, the [http://conal.net/papers/functional-images/ Pan library for functional images] and the reactive library for functional reactive programming (FRP) {{noteref}} make key use of it; his blog also contains [http://conal.net/blog/tag/applicative-functor many examples of <code>Applicative</code> in action]. Building on the work of McBride and Paterson, Elliott also built the [[TypeCompose]] library, which embodies the observation (among others) that <code>Applicative</code> types are closed under composition; therefore, <code>Applicative</code> instances can often be automatically derived for complex types built out of simpler ones.<br />
<br />
Although the [http://hackage.haskell.org/package/parsec Parsec parsing library] ([http://legacy.cs.uu.nl/daan/download/papers/parsec-paper.pdf paper]) was originally designed for use as a monad, in its most common use cases an <code>Applicative</code> instance can be used to great effect; [http://www.serpentine.com/blog/2008/02/06/the-basics-of-applicative-functors-put-to-practical-work/ Bryan O’Sullivan’s blog post] is a good starting point. If the extra power provided by <code>Monad</code> isn’t needed, it’s usually a good idea to use <code>Applicative</code> instead.<br />
<br />
A couple other nice examples of <code>Applicative</code> in action include the [http://chrisdone.com/blog/html/2009-02-10-applicative-configfile-hsql.html ConfigFile and HSQL libraries] and the [http://groups.inf.ed.ac.uk/links/formlets/ formlets library].<br />
<br />
=Monad=<br />
<br />
It’s a safe bet that if you’re reading this, you’ve heard of monads—although it’s quite possible you’ve never heard of <code>Applicative</code> before, or <code>Arrow</code>, or even <code>Monoid</code>. Why are monads such a big deal in Haskell? There are several reasons.<br />
<br />
* Haskell does, in fact, single out monads for special attention by making them the framework in which to construct I/O operations.<br />
* Haskell also singles out monads for special attention by providing a special syntactic sugar for monadic expressions: the <code>do</code>-notation.<br />
* <code>Monad</code> has been around longer than other abstract models of computation such as <code>Applicative</code> or <code>Arrow</code>.<br />
* The more monad tutorials there are, the harder people think monads must be, and the more new monad tutorials are written by people who think they finally “get” monads (the [http://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/ monad tutorial fallacy]).<br />
<br />
I will let you judge for yourself whether these are good reasons.<br />
<br />
In the end, despite all the hoopla, <code>Monad</code> is just another type class. Let’s take a look at its definition.<br />
<br />
==Definition==<br />
<br />
The type class declaration for [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#t:Monad <code>Monad</code>] is:<br />
<br />
<haskell><br />
class Monad m where<br />
return :: a -> m a<br />
(>>=) :: m a -> (a -> m b) -> m b<br />
(>>) :: m a -> m b -> m b<br />
m >> n = m >>= \_ -> n<br />
<br />
fail :: String -> m a<br />
</haskell><br />
<br />
The <code>Monad</code> type class is exported by the <code>Prelude</code>, along with a few standard instances. However, many utility functions are found in [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad.html <code>Control.Monad</code>], and there are also several instances (such as <code>((->) e)</code>) defined in [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-Instances.html <code>Control.Monad.Instances</code>].<br />
<br />
Let’s examine the methods in the <code>Monad</code> class one by one. The type of <code>return</code> should look familiar; it’s the same as <code>pure</code>. Indeed, <code>return</code> ''is'' <code>pure</code>, but with an unfortunate name. (Unfortunate, since someone coming from an imperative programming background might think that <code>return</code> is like the C or Java keyword of the same name, when in fact the similarities are minimal.) From a mathematical point of view, every monad is an applicative functor, but for historical reasons, the <code>Monad</code> type class declaration unfortunately does not require this.<br />
<br />
We can see that <code>(>>)</code> is a specialized version of <code>(>>=)</code>, with a default implementation given. It is only included in the type class declaration so that specific instances of <code>Monad</code> can override the default implementation of <code>(>>)</code> with a more efficient one, if desired. Also, note that although <code>_ >> n = n</code> would be a type-correct implementation of <code>(>>)</code>, it would not correspond to the intended semantics: the intention is that <code>m >> n</code> ignores the ''result'' of <code>m</code>, but not its ''effects''.<br />
<br />
The <code>fail</code> function is an awful hack that has no place in the <code>Monad</code> class; more on this later.<br />
<br />
The only really interesting thing to look at—and what makes <code>Monad</code> strictly more powerful than <code>Applicative</code>—is <code>(>>=)</code>, which is often called ''bind''. An alternative definition of <code>Monad</code> could look like:<br />
<br />
<haskell><br />
class Applicative m => Monad' m where<br />
(>>=) :: m a -> (a -> m b) -> m b<br />
</haskell><br />
<br />
We could spend a while talking about the intuition behind <code>(>>=)</code>—and we will. But first, let’s look at some examples.<br />
<br />
==Instances==<br />
<br />
Even if you don’t understand the intuition behind the <code>Monad</code> class, you can still create instances of it by just seeing where the types lead you. You may be surprised to find that this actually gets you a long way towards understanding the intuition; at the very least, it will give you some concrete examples to play with as you read more about the <code>Monad</code> class in general. The first few examples are from the standard <code>Prelude</code>; the remaining examples are from the [http://hackage.haskell.org/package/transformers <code>transformers</code> package].<br />
<br />
<ul><br />
<li>The simplest possible instance of <code>Monad</code> is [http://hackage.haskell.org/packages/archive/mtl/1.1.0.2/doc/html/Control-Monad-Identity.html <code>Identity</code>], which is described in Dan Piponi’s highly recommended blog post on [http://blog.sigfpe.com/2007/04/trivial-monad.html The Trivial Monad]. Despite being “trivial”, it is a great introduction to the <code>Monad</code> type class, and contains some good exercises to get your brain working.<br />
</li><br />
<li>The next simplest instance of <code>Monad</code> is <code>Maybe</code>. We already know how to write <code>return</code>/<code>pure</code> for <code>Maybe</code>. So how do we write <code>(>>=)</code>? Well, let’s think about its type. Specializing for <code>Maybe</code>, we have<br />
<br />
<haskell><br />
(>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b.<br />
</haskell><br />
<br />
If the first argument to <code>(>>=)</code> is <code>Just x</code>, then we have something of type <code>a</code> (namely, <code>x</code>), to which we can apply the second argument—resulting in a <code>Maybe b</code>, which is exactly what we wanted. What if the first argument to <code>(>>=)</code> is <code>Nothing</code>? In that case, we don’t have anything to which we can apply the <code>a -> Maybe b</code> function, so there’s only one thing we can do: yield <code>Nothing</code>. This instance is:<br />
<br />
<haskell><br />
instance Monad Maybe where<br />
return = Just<br />
(Just x) >>= g = g x<br />
Nothing >>= _ = Nothing<br />
</haskell><br />
<br />
We can already get a bit of intuition as to what is going on here: if we build up a computation by chaining together a bunch of functions with <code>(>>=)</code>, as soon as any one of them fails, the entire computation will fail (because <code>Nothing >>= f</code> is <code>Nothing</code>, no matter what <code>f</code> is). The entire computation succeeds only if all the constituent functions individually succeed. So the <code>Maybe</code> monad models computations which may fail.<br />
</li><br />
<br />
<li>The <code>Monad</code> instance for the list constructor <code>[]</code> is similar to its <code>Applicative</code> instance; see the exercise below.<br />
</li><br />
<br />
<li>Of course, the <code>IO</code> constructor is famously a <code>Monad</code>, but its implementation is somewhat magical, and may in fact differ from compiler to compiler. It is worth emphasizing that the <code>IO</code> monad is the ''only'' monad which is magical. It allows us to build up, in an entirely pure way, values representing possibly effectful computations. The special value <code>main</code>, of type <code>IO ()</code>, is taken by the runtime and actually executed, producing actual effects. Every other monad is functionally pure, and requires no special compiler support. We often speak of monadic values as “effectful computations”, but this is because some monads allow us to write code ''as if'' it has side effects, when in fact the monad is hiding the plumbing which allows these apparent side effects to be implemented in a functionally pure way.<br />
</li><br />
<br />
<li>As mentioned earlier, <code>((->) e)</code> is known as the ''reader monad'', since it describes computations in which a value of type <code>e</code> is available as a read-only environment.<br />
<br />
The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Reader.html <code>Control.Monad.Reader</code>] module provides the <code>Reader e a</code> type, which is just a convenient <code>newtype</code> wrapper around <code>(e -> a)</code>, along with an appropriate <code>Monad</code> instance and some <code>Reader</code>-specific utility functions such as <code>ask</code> (retrieve the environment), <code>asks</code> (retrieve a function of the environment), and <code>local</code> (run a subcomputation under a different environment).<br />
</li><br />
<br />
<li>The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Writer-Lazy.html <code>Control.Monad.Writer</code>] module provides the <code>Writer</code> monad, which allows information to be collected as a computation progresses. <code>Writer w a</code> is isomorphic to <code>(a,w)</code>, where the output value <code>a</code> is carried along with an annotation or “log” of type <code>w</code>, which must be an instance of <code>Monoid</code> (see [[#Monoid|section Monoid]]); the special function <code>tell</code> performs logging.<br />
</li><br />
<br />
<li>The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-State-Lazy.html <code>Control.Monad.State</code>] module provides the <code>State s a</code> type, a <code>newtype</code> wrapper around <code>s -> (a,s)</code>. Something of type <code>State s a</code> represents a stateful computation which produces an <code>a</code> but can access and modify the state of type <code>s</code> along the way. The module also provides <code>State</code>-specific utility functions such as <code>get</code> (read the current state), <code>gets</code> (read a function of the current state), <code>put</code> (overwrite the state), and <code>modify</code> (apply a function to the state).<br />
</li><br />
<br />
<li>The [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Cont.html <code>Control.Monad.Cont</code>] module provides the <code>Cont</code> monad, which represents computations in continuation-passing style. It can be used to suspend and resume computations, and to implement non-local transfers of control, co-routines, other complex control structures—all in a functionally pure way. <code>Cont</code> has been called the [http://blog.sigfpe.com/2008/12/mother-of-all-monads.html “mother of all monads”] because of its universal properties.<br />
</li><br />
</ul><br />
<br />
{{Exercises|<br />
<ol><br />
<li>Implement a <code>Monad</code> instance for the list constructor, <code>[]</code>. Follow the types!</li><br />
<li>Implement a <code>Monad</code> instance for <code>((->) e)</code>.</li><br />
<li>Implement <code>Functor</code> and <code>Monad</code> instances for <code>Free f</code>, defined as<br />
<haskell><br />
data Free f a = Var a<br />
| Node (f (Free f a))<br />
</haskell><br />
You may assume that <code>f</code> has a <code>Functor</code> instance. This is known as the ''free monad'' built from the functor <code>f</code>.<br />
</li><br />
</ol><br />
}}<br />
<br />
==Intuition==<br />
<br />
Let’s look more closely at the type of <code>(>>=)</code>. The basic intuition is that it combines two computations into one larger computation. The first argument, <code>m a</code>, is the first computation. However, it would be boring if the second argument were just an <code>m b</code>; then there would be no way for the computations to interact with one another (actually, this is exactly the situation with <code>Applicative</code>). So, the second argument to <code>(>>=)</code> has type <code>a -> m b</code>: a function of this type, given a ''result'' of the first computation, can produce a second computation to be run. In other words, <code>x >>= k</code> is a computation which runs <code>x</code>, and then uses the result(s) of <code>x</code> to ''decide'' what computation to run second, using the output of the second computation as the result of the entire computation.<br />
<br />
{{note|Actually, because Haskell allows general recursion, this is a lie: using a Haskell parsing library one can recursively construct ''infinite'' grammars, and hence <code>Alternative</code> by itself is enough to parse any context-sensitive language with a finite alphabet. See [http://byorgey.wordpress.com/2012/01/05/parsing-context-sensitive-languages-with-applicative/ Parsing context-sensitive languages with Applicative].}}<br />
Intuitively, it is this ability to use the output from previous computations to decide what computations to run next that makes <code>Monad</code> more powerful than <code>Applicative</code>. The structure of an <code>Applicative</code> computation is fixed, whereas the structure of a <code>Monad</code> computation can change based on intermediate results. This also means that parsers built using an <code>Applicative</code> interface can only parse context-free languages; in order to parse context-sensitive languages a <code>Monad</code> interface is needed.{{noteref}}<br />
<br />
To see the increased power of <code>Monad</code> from a different point of view, let’s see what happens if we try to implement <code>(>>=)</code> in terms of <code>fmap</code>, <code>pure</code>, and <code>(<*>)</code>. We are given a value <code>x</code> of type <code>m a</code>, and a function <code>k</code> of type <code>a -> m b</code>, so the only thing we can do is apply <code>k</code> to <code>x</code>. We can’t apply it directly, of course; we have to use <code>fmap</code> to lift it over the <code>m</code>. But what is the type of <code>fmap k</code>? Well, it’s <code>m a -> m (m b)</code>. So after we apply it to <code>x</code>, we are left with something of type <code>m (m b)</code>—but now we are stuck; what we really want is an <code>m b</code>, but there’s no way to get there from here. We can ''add'' <code>m</code>’s using <code>pure</code>, but we have no way to ''collapse'' multiple <code>m</code>’s into one.<br />
<br />
{{note|1=You might hear some people claim that that the definition in terms of <code>return</code>, <code>fmap</code>, and <code>join</code> is the “math definition” and the definition in terms of <code>return</code> and <code>(>>=)</code> is something specific to Haskell. In fact, both definitions were known in the mathematics community long before Haskell picked up monads.}}<br />
<br />
This ability to collapse multiple <code>m</code>’s is exactly the ability provided by the function <code>join :: m (m a) -> m a</code>, and it should come as no surprise that an alternative definition of <code>Monad</code> can be given in terms of <code>join</code>:<br />
<br />
<haskell><br />
class Applicative m => Monad'' m where<br />
join :: m (m a) -> m a<br />
</haskell><br />
<br />
In fact, the canonical definition of monads in category theory is in terms of <code>return</code>, <code>fmap</code>, and <code>join</code> (often called <math>\eta</math>, <math>T</math>, and <math>\mu</math> in the mathematical literature). Haskell uses an alternative formulation with <code>(>>=)</code> instead of <code>join</code> since it is more convenient to use {{noteref}}. However, sometimes it can be easier to think about <code>Monad</code> instances in terms of <code>join</code>, since it is a more “atomic” operation. (For example, <code>join</code> for the list monad is just <code>concat</code>.)<br />
<br />
{{Exercises|<br />
# Implement <code>(>>{{=}})</code> in terms of <code>fmap</code> (or <code>liftM</code>) and <code>join</code>.<br />
# Now implement <code>join</code> and <code>fmap</code> (<code>liftM</code>) in terms of <code>(>>{{=}})</code> and <code>return</code>.<br />
}}<br />
<br />
==Utility functions==<br />
<br />
The [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad.html <code>Control.Monad</code>] module provides a large number of convenient utility functions, all of which can be implemented in terms of the basic <code>Monad</code> operations (<code>return</code> and <code>(>>=)</code> in particular). We have already seen one of them, namely, <code>join</code>. We also mention some other noteworthy ones here; implementing these utility functions oneself is a good exercise. For a more detailed guide to these functions, with commentary and example code, see Henk-Jan van Tuyl’s [http://members.chello.nl/hjgtuyl/tourdemonad.html tour].<br />
<br />
{{note|Still, it is unclear how this "bug" should be fixed. Making <code>Monad</code> require a <code>Functor</code> instance has some drawbacks, as mentioned in this [http://www.haskell.org/pipermail/haskell-prime/2011-January/003312.html 2011 mailing-list discussion]. —Geheimdienst}}<br />
<br />
* <code>liftM :: Monad m => (a -> b) -> m a -> m b</code>. This should be familiar; of course, it is just <code>fmap</code>. The fact that we have both <code>fmap</code> and <code>liftM</code> is an unfortunate consequence of the fact that the <code>Monad</code> type class does not require a <code>Functor</code> instance, even though mathematically speaking, every monad is a functor. However, <code>fmap</code> and <code>liftM</code> are essentially interchangeable, since it is a bug (in a social rather than technical sense) for any type to be an instance of <code>Monad</code> without also being an instance of <code>Functor</code> {{noteref}}.<br />
<br />
* <code>ap :: Monad m => m (a -> b) -> m a -> m b</code> should also be familiar: it is equivalent to <code>(<*>)</code>, justifying the claim that the <code>Monad</code> interface is strictly more powerful than <code>Applicative</code>. We can make any <code>Monad</code> into an instance of <code>Applicative</code> by setting <code>pure = return</code> and <code>(<*>) = ap</code>.<br />
<br />
* <code>sequence :: Monad m => [m a] -> m [a]</code> takes a list of computations and combines them into one computation which collects a list of their results. It is again something of a historical accident that <code>sequence</code> has a <code>Monad</code> constraint, since it can actually be implemented only in terms of <code>Applicative</code>. There is an additional generalization of <code>sequence</code> to structures other than lists, which will be discussed in the [[#Traversable|section on <code>Traversable</code>]].<br />
<br />
* <code>replicateM :: Monad m => Int -> m a -> m [a]</code> is simply a combination of [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html#v:replicate <code>replicate</code>] and <code>sequence</code>.<br />
<br />
* <code>when :: Monad m => Bool -> m () -> m ()</code> conditionally executes a computation, evaluating to its second argument if the test is <code>True</code>, and to <code>return ()</code> if the test is <code>False</code>. A collection of other sorts of monadic conditionals can be found in the [http://hackage.haskell.org/package/IfElse <code>IfElse</code> package].<br />
<br />
* <code>mapM :: Monad m => (a -> m b) -> [a] -> m [b]</code> maps its first argument over the second, and <code>sequence</code>s the results. The <code>forM</code> function is just <code>mapM</code> with its arguments reversed; it is called <code>forM</code> since it models generalized <code>for</code> loops: the list <code>[a]</code> provides the loop indices, and the function <code>a -> m b</code> specifies the “body” of the loop for each index.<br />
<br />
* <code>(=<<) :: Monad m => (a -> m b) -> m a -> m b</code> is just <code>(>>=)</code> with its arguments reversed; sometimes this direction is more convenient since it corresponds more closely to function application.<br />
<br />
* <code>(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c</code> is sort of like function composition, but with an extra <code>m</code> on the result type of each function, and the arguments swapped. We’ll have more to say about this operation later. There is also a flipped variant, <code>(<=<)</code>.<br />
<br />
* The <code>guard</code> function is for use with instances of <code>MonadPlus</code>, which is discussed at the end of the [[#Monoid|<code>Monoid</code> section]].<br />
<br />
Many of these functions also have “underscored” variants, such as <code>sequence_</code> and <code>mapM_</code>; these variants throw away the results of the computations passed to them as arguments, using them only for their side effects.<br />
<br />
Other monadic functions which are occasionally useful include <code>filterM</code>, <code>zipWithM</code>, <code>foldM</code>, and <code>forever</code>. <br />
<br />
==Laws==<br />
<br />
There are several laws that instances of <code>Monad</code> should satisfy (see also the [[Monad laws]] wiki page). The standard presentation is:<br />
<br />
<haskell><br />
return a >>= k = k a<br />
m >>= return = m<br />
m >>= (\x -> k x >>= h) = (m >>= k) >>= h<br />
<br />
fmap f xs = xs >>= return . f = liftM f xs<br />
</haskell><br />
<br />
The first and second laws express the fact that <code>return</code> behaves nicely: if we inject a value <code>a</code> into a monadic context with <code>return</code>, and then bind to <code>k</code>, it is the same as just applying <code>k</code> to <code>a</code> in the first place; if we bind a computation <code>m</code> to <code>return</code>, nothing changes. The third law essentially says that <code>(>>=)</code> is associative, sort of. The last law ensures that <code>fmap</code> and <code>liftM</code> are the same for types which are instances of both <code>Functor</code> and <code>Monad</code>—which, as already noted, should be every instance of <code>Monad</code>.<br />
<br />
{{note|I like to pronounce this operator “fish”.}}<br />
<br />
However, the presentation of the above laws, especially the third, is marred by the asymmetry of <code>(>>=)</code>. It’s hard to look at the laws and see what they’re really saying. I prefer a much more elegant version of the laws, which is formulated in terms of <code>(>=>)</code> {{noteref}}. Recall that <code>(>=>)</code> “composes” two functions of type <code>a -> m b</code> and <code>b -> m c</code>. You can think of something of type <code>a -> m b</code> (roughly) as a function from <code>a</code> to <code>b</code> which may also have some sort of effect in the context corresponding to <code>m</code>. <code>(>=>)</code> lets us compose these “effectful functions”, and we would like to know what properties <code>(>=>)</code> has. The monad laws reformulated in terms of <code>(>=>)</code> are:<br />
<br />
<haskell><br />
return >=> g = g<br />
g >=> return = g<br />
(g >=> h) >=> k = g >=> (h >=> k)<br />
</haskell><br />
<br />
{{note|As fans of category theory will note, these laws say precisely that functions of type <code>a -> m b</code> are the arrows of a category with <code>(>{{=}}>)</code> as composition! Indeed, this is known as the ''Kleisli category'' of the monad <code>m</code>. It will come up again when we discuss <code>Arrow</code>s.}}<br />
<br />
Ah, much better! The laws simply state that <code>return</code> is the identity of <code>(>=>)</code>, and that <code>(>=>)</code> is associative {{noteref}}.<br />
<br />
There is also a formulation of the monad laws in terms of <code>fmap</code>, <code>return</code>, and <code>join</code>; for a discussion of this formulation, see the Haskell [http://en.wikibooks.org/wiki/Haskell/Category_theory wikibook page on category theory].<br />
<br />
{{Exercises|<br />
# Given the definition <code>g >{{=}}> h {{=}} \x -> g x >>{{=}} h</code>, prove the equivalence of the above laws and the usual monad laws.<br />
}}<br />
<br />
==<code>do</code> notation==<br />
<br />
Haskell’s special <code>do</code> notation supports an “imperative style” of programming by providing syntactic sugar for chains of monadic expressions. The genesis of the notation lies in realizing that something like <code>a >>= \x -> b >> c >>= \y -> d </code> can be more readably written by putting successive computations on separate lines:<br />
<br />
<haskell><br />
a >>= \x -><br />
b >><br />
c >>= \y -><br />
d<br />
</haskell><br />
<br />
This emphasizes that the overall computation consists of four computations <code>a</code>, <code>b</code>, <code>c</code>, and <code>d</code>, and that <code>x</code> is bound to the result of <code>a</code>, and <code>y</code> is bound to the result of <code>c</code> (<code>b</code>, <code>c</code>, and <code>d</code> are allowed to refer to <code>x</code>, and <code>d</code> is allowed to refer to <code>y</code> as well). From here it is not hard to imagine a nicer notation:<br />
<br />
<haskell><br />
do { x <- a<br />
; b<br />
; y <- c<br />
; d<br />
}<br />
</haskell><br />
<br />
(The curly braces and semicolons may optionally be omitted; the Haskell parser uses layout to determine where they should be inserted.) This discussion should make clear that <code>do</code> notation is just syntactic sugar. In fact, <code>do</code> blocks are recursively translated into monad operations (almost) like this:<br />
<br />
<pre><br />
do e → e<br />
do { e; stmts } → e >> do { stmts }<br />
do { v <- e; stmts } → e >>= \v -> do { stmts }<br />
do { let decls; stmts} → let decls in do { stmts }<br />
</pre><br />
<br />
This is not quite the whole story, since <code>v</code> might be a pattern instead of a variable. For example, one can write<br />
<br />
<haskell><br />
do (x:xs) <- foo<br />
bar x<br />
</haskell><br />
<br />
but what happens if <code>foo</code> produces an empty list? Well, remember that ugly <code>fail</code> function in the <code>Monad</code> type class declaration? That’s what happens. See [http://haskell.org/onlinereport/exps.html#sect3.14 section 3.14 of the Haskell Report] for the full details. See also the discussion of <code>MonadPlus</code> and <code>MonadZero</code> in the [[#Other monoidal classes: Alternative, MonadPlus, ArrowPlus|section on other monoidal classes]].<br />
<br />
A final note on intuition: <code>do</code> notation plays very strongly to the “computational context” point of view rather than the “container” point of view, since the binding notation <code>x <- m</code> is suggestive of “extracting” a single <code>x</code> from <code>m</code> and doing something with it. But <code>m</code> may represent some sort of a container, such as a list or a tree; the meaning of <code>x <- m</code> is entirely dependent on the implementation of <code>(>>=)</code>. For example, if <code>m</code> is a list, <code>x <- m</code> actually means that <code>x</code> will take on each value from the list in turn.<br />
<br />
==Further reading==<br />
<br />
Philip Wadler was the first to propose using monads to structure functional programs. [http://homepages.inf.ed.ac.uk/wadler/topics/monads.html His paper] is still a readable introduction to the subject.<br />
<br />
{{note|1=<br />
[[All About Monads]],<br />
[http://haskell.org/haskellwiki/Monads_as_Containers Monads as containers],<br />
[http://en.wikibooks.org/w/index.php?title=Haskell/Understanding_monads&oldid=933545 Understanding monads],<br />
[[The Monadic Way]],<br />
[http://blog.sigfpe.com/2006/08/you-could-have-invented-monads-and.html You Could Have Invented Monads! (And Maybe You Already Have.)],<br />
[http://www.haskell.org/pipermail/haskell-cafe/2006-November/019190.html there’s a monster in my Haskell!],<br />
[http://kawagner.blogspot.com/2007/02/understanding-monads-for-real.html Understanding Monads. For real.],<br />
[http://www.randomhacks.net/articles/2007/03/12/monads-in-15-minutes Monads in 15 minutes: Backtracking and Maybe],<br />
[http://haskell.org/haskellwiki/Monads_as_computation Monads as computation],<br />
[http://metafoo.co.uk/practical-monads.txt Practical Monads]}}<br />
<br />
There are, of course, numerous monad tutorials of varying quality {{noteref}}.<br />
<br />
A few of the best include Cale Gibbard’s [http://haskell.org/haskellwiki/Monads_as_Containers Monads as containers] and [http://haskell.org/haskellwiki/Monads_as_computation Monads as computation]; Jeff Newbern’s [[All About Monads]], a comprehensive guide with lots of examples; and Dan Piponi’s [http://blog.sigfpe.com/2006/08/you-could-have-invented-monads-and.html You Could Have Invented Monads!], which features great exercises. If you just want to know how to use <code>IO</code>, you could consult the [[Introduction to IO]]. Even this is just a sampling; the [[monad tutorials timeline]] is a more complete list. (All these monad tutorials have prompted parodies like [http://koweycode.blogspot.com/2007/01/think-of-monad.html think of a monad ...] as well as other kinds of backlash like [http://ahamsandwich.wordpress.com/2007/07/26/monads-and-why-monad-tutorials-are-all-awful/ Monads! (and Why Monad Tutorials Are All Awful)] or [http://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/ Abstraction, intuition, and the “monad tutorial fallacy”].)<br />
<br />
Other good monad references which are not necessarily tutorials include [http://members.chello.nl/hjgtuyl/tourdemonad.html Henk-Jan van Tuyl’s tour] of the functions in <code>Control.Monad</code>, Dan Piponi’s [http://blog.sigfpe.com/2006/10/monads-field-guide.html field guide], Tim Newsham’s [http://www.thenewsh.com/~newsham/haskell/monad.html What’s a Monad?], and Chris Smith's excellent article [http://cdsmith.wordpress.com/2012/04/18/why-do-monads-matter/ Why Do Monads Matter?]. There are also many blog posts which have been written on various aspects of monads; a collection of links can be found under [[Blog articles/Monads]].<br />
<br />
For help constructing monads from scratch, and for obtaining a "deep embedding" of monad operations suitable for use in, say, compiling a domain-specific language, see [http://projects.haskell.org/operational apfelmus's operational package].<br />
<br />
One of the quirks of the <code>Monad</code> class and the Haskell type system is that it is not possible to straightforwardly declare <code>Monad</code> instances for types which require a class constraint on their data, even if they are monads from a mathematical point of view. For example, <code>Data.Set</code> requires an <code>Ord</code> constraint on its data, so it cannot be easily made an instance of <code>Monad</code>. A solution to this problem was [http://www.randomhacks.net/articles/2007/03/15/data-set-monad-haskell-macros first described by Eric Kidd], and later made into a [http://hackage.haskell.org/cgi-bin/hackage-scripts/package/rmonad library named rmonad] by Ganesh Sittampalam and Peter Gavin.<br />
<br />
There are many good reasons for eschewing <code>do</code> notation; some have gone so far as to [[Do_notation_considered_harmful|consider it harmful]].<br />
<br />
Monads can be generalized in various ways; for an exposition of one possibility, see Robert Atkey’s paper on [http://homepages.inf.ed.ac.uk/ratkey/paramnotions-jfp.pdf parameterized monads], or Dan Piponi’s [http://blog.sigfpe.com/2009/02/beyond-monads.html Beyond Monads].<br />
<br />
For the categorically inclined, monads can be viewed as monoids ([http://blog.sigfpe.com/2008/11/from-monoids-to-monads.html From Monoids to Monads]) and also as closure operators [http://blog.plover.com/math/monad-closure.html Triples and Closure]. Derek Elkins’s article in [http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf issue 13 of the Monad.Reader] contains an exposition of the category-theoretic underpinnings of some of the standard <code>Monad</code> instances, such as <code>State</code> and <code>Cont</code>. Jonathan Hill and Keith Clarke have [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.6497 an early paper explaining the connection between monads as they arise in category theory and as used in functional programming]. There is also a [http://okmij.org/ftp/Computation/IO-monad-history.html web page by Oleg Kiselyov] explaining the history of the IO monad.<br />
<br />
Links to many more research papers related to monads can be found under [[Research papers/Monads and arrows]].<br />
<br />
=Monad transformers=<br />
<br />
One would often like to be able to combine two monads into one: for example, to have stateful, nondeterministic computations (<code>State</code> + <code>[]</code>), or computations which may fail and can consult a read-only environment (<code>Maybe</code> + <code>Reader</code>), and so on. Unfortunately, monads do not compose as nicely as applicative functors (yet another reason to use <code>Applicative</code> if you don’t need the full power that <code>Monad</code> provides), but some monads can be combined in certain ways.<br />
<br />
==Standard monad transformers==<br />
<br />
The [http://hackage.haskell.org/package/transformers transformers] library provides a number of standard ''monad transformers''. Each monad transformer adds a particular capability/feature/effect to any existing monad.<br />
<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Identity.html <code>IdentityT</code>] is the identity transformer, which maps a monad to (something isomorphic to) itself. This may seem useless at first glance, but it is useful for the same reason that the <code>id</code> function is useful -- it can be passed as an argument to things which are parameterized over an arbitrary monad transformer, when you do not actually want any extra capabilities.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-State.html <code>StateT</code>] adds a read-write state.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Reader.html <code>ReaderT</code>] adds a read-only environment.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Writer.html <code>WriterT</code>] adds a write-only log.<br />
* [http://hackage.haskell.org/packages/archive/transformers/0.2.2.0/doc/html/Control-Monad-Trans-RWS.html <code>RWST</code>] conveniently combines <code>ReaderT</code>, <code>WriterT</code>, and <code>StateT</code> into one.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Maybe.html <code>MaybeT</code>] adds the possibility of failure.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Error.html <code>ErrorT</code>] adds the possibility of failure with an arbitrary type to represent errors.<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-List.html <code>ListT</code>] adds non-determinism (however, see the discussion of <code>ListT</code> below).<br />
* [http://hackage.haskell.org/packages/archive/transformers/latest/doc/html/Control-Monad-Trans-Cont.html <code>ContT</code>] adds continuation handling.<br />
<br />
For example, <code>StateT s Maybe</code> is an instance of <code>Monad</code>; computations of type <code>StateT s Maybe a</code> may fail, and have access to a mutable state of type <code>s</code>. Monad transformers can be multiply stacked. One thing to keep in mind while using monad transformers is that the order of composition matters. For example, when a <code>StateT s Maybe a</code> computation fails, the state ceases being updated (indeed, it simply disappears); on the other hand, the state of a <code>MaybeT (State s) a</code> computation may continue to be modified even after the computation has "failed". This may seem backwards, but it is correct. Monad transformers build composite monads “inside out”; <code>MaybeT (State s) a</code> is isomorphic to <code>s -> (Maybe a, s)</code>. (Lambdabot has an indispensable <code>@unmtl</code> command which you can use to “unpack” a monad transformer stack in this way.)<br />
Intuitively, the monads become "more fundamental" the further down in the stack you get, and the effects of a given monad "have precedence" over the effects of monads further up the stack. Of course, this is just handwaving, and if you are unsure of the proper order for some monads you wish to combine, there is no substitute for using <code>@unmtl</code> or simply trying out the various options.<br />
<br />
==Definition and laws==<br />
<br />
All monad transformers should implement the <code>MonadTrans</code> type class, defined in <code>Control.Monad.Trans.Class</code>:<br />
<br />
<haskell><br />
class MonadTrans t where<br />
lift :: Monad m => m a -> t m a<br />
</haskell><br />
<br />
It allows arbitrary computations in the base monad <code>m</code> to be “lifted” into computations in the transformed monad <code>t m</code>. (Note that type application associates to the left, just like function application, so <code>t m a = (t m) a</code>.)<br />
<br />
<code>lift</code> must satisfy the laws<br />
<haskell><br />
lift . return = return<br />
lift (m >>= f) = lift m >>= (lift . f)<br />
</haskell><br />
which intuitively state that <code>lift</code> transforms <code>m a</code> computations into <code>t m a</code> computations in a "sensible" way, which sends the <code>return</code> and <code>(>>=)</code> of <code>m</code> to the <code>return</code> and <code>(>>=)</code> of <code>t m</code>.<br />
<br />
{{Exercises|<br />
# What is the kind of <code>t</code> in the declaration of <code>MonadTrans</code>?<br />
}}<br />
<br />
==Transformer type classes and "capability" style==<br />
<br />
{{note|The only problem with this scheme is the quadratic number of instances required as the number of standard monad transformers grows—but as the current set of standard monad transformers seems adequate for most common use cases, this may not be that big of a deal.}}<br />
<br />
There are also type classes (provided by the [http://hackage.haskell.org/package/mtl <code>mtl</code> package]) for the operations of each transformer. For example, the <code>MonadState</code> type class provides the state-specific methods <code>get</code> and <code>put</code>, allowing you to conveniently use these methods not only with <code>State</code>, but with any monad which is an instance of <code>MonadState</code>—including <code>MaybeT (State s)</code>, <code>StateT s (ReaderT r IO)</code>, and so on. Similar type classes exist for <code>Reader</code>, <code>Writer</code>, <code>Cont</code>, <code>IO</code>, and others {{noteref}}.<br />
<br />
These type classes serve two purposes. First, they get rid of (most of) the need for explicitly using <code>lift</code>, giving a type-directed way to automatically determine the right number of calls to <code>lift</code>. Simply writing <code>put</code> will be automatically translated into <code>lift . put</code>, <code>lift . lift . put</code>, or something similar depending on what concrete monad stack you are using.<br />
<br />
Second, they give you more flexibility to switch between different concrete monad stacks. For example, if you are writing a state-based algorithm, don't write<br />
<haskell><br />
foo :: State Int Char<br />
foo = modify (*2) >> return 'x'<br />
</haskell><br />
but rather<br />
<haskell><br />
foo :: MonadState Int m => m Char<br />
foo = modify (*2) >> return 'x'<br />
</haskell><br />
Now, if somewhere down the line you realize you need to introduce the possibility of failure, you might switch from <code>State Int</code> to <code>MaybeT (State Int)</code>. The type of the first version of <code>foo</code> would need to be modified to reflect this change, but the second version of <code>foo</code> can still be used as-is.<br />
<br />
However, this sort of "capability-based" style (<i>e.g.</i> specifying that <code>foo</code> works for any monad with the "state capability") quickly runs into problems when you try to naively scale it up: for example, what if you need to maintain two independent states? A framework for solving this and related problems is described by Schrijvers and Olivera ([http://users.ugent.be/~tschrijv/Research/papers/icfp2011.pdf Monads, zippers and views: virtualizing the monad stack, ICFP 2011]) and is implemented in the [http://hackage.haskell.org/package/Monatron <code>Monatron</code> package].<br />
<br />
==Composing monads==<br />
<br />
Is the composition of two monads always a monad? As hinted previously, the answer is no. For example, ''XXX insert example here''.<br />
<br />
Since <code>Applicative</code> functors are closed under composition, the problem must lie with <code>join</code>. Indeed, suppose <code>m</code> and <code>n</code> are arbitrary monads; to make a monad out of their composition we would need to be able to implement<br />
<haskell><br />
join :: m (n (m (n a))) -> m (n a)<br />
</haskell><br />
but it is not clear how this could be done in general. The <code>join</code> method for <code>m</code> is no help, because the two occurrences of <code>m</code> are not next to each other (and likewise for <code>n</code>).<br />
<br />
However, one situation in which it can be done is if <code>n</code> ''distributes'' over <code>m</code>, that is, if there is a function<br />
<haskell><br />
distrib :: n (m a) -> m (n a)<br />
</haskell><br />
satisfying certain laws. See Jones and Duponcheel ([http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.2605 Composing Monads]); see also the [[#Traversable|section on Traversable]].<br />
<br />
{{Exercises|<br />
* Implement <code>join :: M (N (M (N a))) -> M (N a)</code>, given <code>distrib :: N (M a) -> M (N a)</code> and assuming <code>M</code> and <code>N</code> are instances of <code>Monad</code>.<br />
}}<br />
<br />
==Further reading==<br />
<br />
Much of the monad transformer library (originally [http://hackage.haskell.org/package/mtl <code>mtl</code>], now split between <code>mtl</code> and [http://hackage.haskell.org/package/transformers <code>transformers</code>]), including the <code>Reader</code>, <code>Writer</code>, <code>State</code>, and other monads, as well as the monad transformer framework itself, was inspired by Mark Jones’s classic paper [http://web.cecs.pdx.edu/~mpj/pubs/springschool.html Functional Programming with Overloading and Higher-Order Polymorphism]. It’s still very much worth a read—and highly readable—after almost fifteen years.<br />
<br />
See [http://article.gmane.org/gmane.comp.lang.haskell.libraries/17139 Edward Kmett's mailing list message] for a description of the history and relationships among monad transformer packages (<code>mtl</code>, <code>transformers</code>, <code>monads-fd</code>, <code>monads-tf</code>).<br />
<br />
There are two excellent references on monad transformers. Martin Grabmüller’s [http://www.grabmueller.de/martin/www/pub/Transformers.en.html Monad Transformers Step by Step] is a thorough description, with running examples, of how to use monad transformers to elegantly build up computations with various effects. [http://cale.yi.org/index.php/How_To_Use_Monad_Transformers Cale Gibbard’s article] on how to use monad transformers is more practical, describing how to structure code using monad transformers to make writing it as painless as possible. Another good starting place for learning about monad transformers is a [http://blog.sigfpe.com/2006/05/grok-haskell-monad-transformers.html blog post by Dan Piponi].<br />
<br />
The <code>ListT</code> transformer from the <code>transformers</code> package comes with the caveat that <code>ListT m</code> is only a monad when <code>m</code> is ''commutative'', that is, when <code>ma >>= \a -> mb >>= \b -> foo</code> is equivalent to <code>mb >>= \b -> ma >>= \a -> foo</code> (i.e. the order of <code>m</code>'s effects does not matter). For one explanation why, see Dan Piponi's blog post [http://blog.sigfpe.com/2006/11/why-isnt-listt-monad.html "Why isn't <code><nowiki>ListT []</nowiki></code> a monad"]. For more examples, as well as a design for a version of <code>ListT</code> which does not have this problem, see [http://haskell.org/haskellwiki/ListT_done_right <code>ListT</code> done right].<br />
<br />
There is an alternative way to compose monads, using coproducts, as described by [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.3581 Lüth and Ghani]. This method is interesting but has not (yet?) seen widespread use.<br />
<br />
=MonadFix=<br />
<br />
''Note: <code>MonadFix</code> is included here for completeness (and because it is interesting) but seems not to be used much. Skipping this section on a first read-through is perfectly OK (and perhaps even recommended).''<br />
<br />
==<code>mdo</code>/<code>do rec</code> notation==<br />
<br />
{{note|In GHC 7.6, the flag has been changed to <code>-XRecursiveDo</code>.}}<br />
The <code>MonadFix</code> class describes monads which support the special fixpoint operation <code>mfix :: (a -> m a) -> m a</code>, which allows the output of monadic computations to be defined via (effectful) recursion. This is [http://www.haskell.org/ghc/docs/latest/html/users_guide/syntax-extns.html#recursive-do-notation supported in GHC] by a special “recursive do” notation, enabled by the <code>-XDoRec</code> flag{{noteref}}. Within a <code>do</code> block, one may have a nested <code>rec</code> block, like so:<br />
<haskell><br />
do { x <- foo<br />
; rec { y <- baz<br />
; z <- bar<br />
; bob<br />
}<br />
; w <- frob<br />
}<br />
</haskell><br />
Normally (if we had <code>do</code> in place of <code>rec</code> in the above example), <code>y</code> would be in scope in <code>bar</code> and <code>bob</code> but not in <code>baz</code>, and <code>z</code> would be in scope only in <code>bob</code>. With the <code>rec</code>, however, <code>y</code> and <code>z</code> are both in scope in all three of <code>baz</code>, <code>bar</code>, and <code>bob</code>. A <code>rec</code> block is analogous to a <code>let</code> block such as<br />
<haskell><br />
let { y = baz<br />
; z = bar<br />
}<br />
in bob<br />
</haskell><br />
because, in Haskell, every variable bound in a <code>let</code>-block is in scope throughout the entire block. (From this point of view, Haskell's normal <code>do</code> blocks are analogous to Scheme's <code>let*</code> construct.)<br />
<br />
What could such a feature be used for? One of the motivating examples given in the original paper describing <code>MonadFix</code> (see below) is encoding circuit descriptions. A line in a <code>do</code>-block such as <br />
<haskell><br />
x <- gate y z<br />
</haskell><br />
describes a gate whose input wires are labeled <code>y</code> and <code>z</code> and whose output wire is labeled <code>x</code>. Many (most?) useful circuits, however, involve some sort of feedback loop, making them impossible to write in a normal <code>do</code>-block (since some wire would have to be mentioned as an input ''before'' being listed as an output). Using a <code>rec</code> block solves this problem.<br />
<br />
==Examples and intuition==<br />
<br />
Of course, not every monad supports such recursive binding. However, as mentioned above, it suffices to have an implementation of <code>mfix :: (a -> m a) -> m a</code>, satisfying a few laws. Let's try implementing <code>mfix</code> for the <code>Maybe</code> monad. That is, we want to implement a function<br />
<haskell><br />
maybeFix :: (a -> Maybe a) -> Maybe a<br />
</haskell><br />
{{note|Actually, <code>fix</code> is implemented slightly differently for efficiency reasons; but the given definition is equivalent and simpler for the present purpose.}}<br />
Let's think for a moment about the implementation {{noteref}} of the non-monadic <code>fix :: (a -> a) -> a</code>:<br />
<haskell><br />
fix f = f (fix f)<br />
</haskell><br />
Inspired by <code>fix</code>, our first attempt at implementing <code>maybeFix</code> might be something like<br />
<haskell><br />
maybeFix :: (a -> Maybe a) -> Maybe a<br />
maybeFix f = maybeFix f >>= f<br />
</haskell><br />
This has the right type. However, something seems wrong: there is nothing in particular here about <code>Maybe</code>; <code>maybeFix</code> actually has the more general type <code>Monad m => (a -> m a) -> m a</code>. But didn't we just say that not all monads support <code>mfix</code>?<br />
<br />
The answer is that although this implementation of <code>maybeFix</code> has the right type, it does ''not'' have the intended semantics. If we think about how <code>(>>=)</code> works for the <code>Maybe</code> monad (by pattern-matching on its first argument to see whether it is <code>Nothing</code> or <code>Just</code>) we can see that this definition of <code>maybeFix</code> is completely useless: it will just recurse infinitely, trying to decide whether it is going to return <code>Nothing</code> or <code>Just</code>, without ever even so much as a glance in the direction of <code>f</code>.<br />
<br />
The trick is to simply ''assume'' that <code>maybeFix</code> will return <code>Just</code>, and get on with life!<br />
<haskell><br />
maybeFix :: (a -> Maybe a) -> Maybe a<br />
maybeFix f = ma<br />
where ma = f (fromJust ma)<br />
</haskell><br />
This says that the result of <code>maybeFix</code> is <code>ma</code>, and assuming that <code>ma = Just x</code>, it is defined (recursively) to be equal to <code>f x</code>.<br />
<br />
Why is this OK? Isn't <code>fromJust</code> almost as bad as <code>unsafePerformIO</code>? Well, usually, yes. This is just about the only situation in which it is justified! The interesting thing to note is that <code>maybeFix</code> ''will never crash'' -- although it may, of course, fail to terminate. The only way we could get a crash is if we try to evaluate <code>fromJust ma</code> when we know that <code>ma = Nothing</code>. But how could we know <code>ma = Nothing</code>? Since <code>ma</code> is defined as <code>f (fromJust ma)</code>, it must be that this expression has already been evaluated to <code>Nothing</code> -- in which case there is no reason for us to be evaluating <code>fromJust ma</code> in the first place! <br />
<br />
To see this from another point of view, we can consider three possibilities. First, if <code>f</code> outputs <code>Nothing</code> without looking at its argument, then <code>maybeFix f</code> clearly returns <code>Nothing</code>. Second, if <code>f</code> always outputs <code>Just x</code>, where <code>x</code> depends on its argument, then the recursion can proceed usefully: <code>fromJust ma</code> will be able to evaluate to <code>x</code>, thus feeding <code>f</code>'s output back to it as input. Third, if <code>f</code> tries to use its argument to decide whether to output <code>Just</code> or <code>Nothing</code>, then <code>maybeFix f</code> will not terminate: evaluating <code>f</code>'s argument requires evaluating <code>ma</code> to see whether it is <code>Just</code>, which requires evaluating <code>f (fromJust ma)</code>, which requires evaluating <code>ma</code>, ... and so on.<br />
<br />
There are also instances of <code>MonadFix</code> for lists (which works analogously to the instance for <code>Maybe</code>), for <code>ST</code>, and for <code>IO</code>. The [http://hackage.haskell.org/packages/archive/base/latest/doc/html/src/System-IO.html#fixIO instance for <code>IO</code>] is particularly amusing: it creates a new <code>IORef</code> (with a dummy value), immediately reads its contents using <code>unsafeInterleaveIO</code> (which delays the actual reading lazily until the value is needed), uses the contents of the <code>IORef</code> to compute a new value, which it then writes back into the <code>IORef</code>. It almost seems, spookily, that <code>mfix</code> is sending a value back in time to itself through the <code>IORef</code> -- though of course what is really going on is that the reading is delayed just long enough (via <code>unsafeInterleaveIO</code>) to get the process bootstrapped.<br />
<br />
{{Exercises|<br />
* Implement a <code>MonadFix</code> instance for <code>[]</code>.<br />
}}<br />
<br />
==GHC 7.6 changes==<br />
<br />
GHC 7.6 reinstated the old <code>mdo</code> syntax, so the example at the start of this section can be written<br />
<br />
<haskell><br />
mdo { x <- foo<br />
; y <- baz<br />
; z <- bar<br />
; bob<br />
; w <- frob<br />
}<br />
</haskell><br />
<br />
which will be translated into the original example (assuming that, say, <code>bar</code> and <code>bob</code> refer to <code>y</code>. The difference is that <code>mdo</code> will analyze the code in order to find minimal recursive blocks, which will be placed in <code>rec</code> blocks, whereas <code>rec</code> blocks desugar directly into calls to <code>mfix</code> without any further analysis. <br />
==Further reading==<br />
<br />
For more information (such as the precise desugaring rules for <code>rec</code> blocks), see Levent Erkök and John Launchbury's 2002 Haskell workshop paper, [http://sites.google.com/site/leventerkok/recdo.pdf?attredirects=0 A Recursive do for Haskell], or for full details, Levent Erkök’s thesis, [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.15.1543&rep=rep1&type=pdf Value Recursion in Monadic Computations]. (Note, while reading, that <code>MonadFix</code> used to be called <code>MonadRec</code>.) You can also read the [http://www.haskell.org/ghc/docs/latest/html/users_guide/syntax-extns.html#recursive-do-notation GHC user manual section on recursive do-notation].<br />
<br />
=Semigroup=<br />
<br />
A semigroup is a set <math>S\ </math> together with a binary operation <math>\oplus\ </math> which<br />
combines elements from <math>S\ </math>. The <math>\oplus\ </math> operator is required to be associative<br />
(that is, <math>(a \oplus b) \oplus c = a \oplus (b \oplus c)\ </math>, for any<br />
<math>a,b,c\ </math> which are elements of <math>S\ </math>).<br />
<br />
For example, the natural numbers under addition form a semigroup: the sum of any two natural numbers is a natural number, and <math>(a+b)+c = a+(b+c)\ </math> for any natural numbers <math>a\ </math>, <math>b\ </math>, and <math>c\,\ </math>. The integers under multiplication also form a semigroup, as do the integers (or rationals, or reals) under <math>\max\ </math> or <math>\min\ </math>, Boolean values under conjunction and disjunction, lists under concatenation, functions from a set to itself under composition ... Semigroups show up all over the place, once you know to look for them.<br />
<br />
==Definition==<br />
<br />
Semigroups are not (yet?) defined in the base package, but the {{HackagePackage|id=semigroups}} package provides a standard definition.<br />
<br />
The definition of the <code>Semigroup</code> type class ([http://hackage.haskell.org/packages/archive/semigroups/latest/doc/html/Data-Semigroup.html haddock]) is as follows:<br />
<br />
<haskell><br />
class Semigroup a where<br />
(<>) :: a -> a -> a<br />
<br />
sconcat :: NonEmpty a -> a<br />
sconcat = sconcat (a :| as) = go a as where<br />
go b (c:cs) = b <> go c cs<br />
go b [] = b<br />
<br />
times1p :: Whole n => n -> a -> a<br />
times1p = ...<br />
</haskell><br />
<br />
The really important method is <code>(<>)</code>, representing the associative binary operation. The other two methods have default implementations in terms of <code>(<>)</code>, and are included in the type class in case some instances can give more efficient implementations than the default. <code>sconcat</code> reduces a nonempty list using <code>(<>)</code>; <code>times1p n</code> is equivalent to (but more efficient than) <code>sconcat . replicate n</code>. See the [http://hackage.haskell.org/packages/archive/semigroups/latest/doc/html/Data-Semigroup.html haddock documentation] for more information on <code>sconcat</code> and <code>times1p</code>.<br />
<br />
==Laws==<br />
<br />
The only law is that <code>(<>)</code> must be associative:<br />
<br />
<haskell><br />
(x <> y) <> z = x <> (y <> z)<br />
</haskell><br />
<br />
More coming soon...<br />
<br />
=Monoid=<br />
<br />
Many semigroups have a special element <math>e</math> for which the binary operation <math>\oplus</math> is the identity, that is, <math>e \oplus x = x \oplus e = x</math> for every element <math>x</math>. Such a semigroup-with-identity-element is called a ''monoid''.<br />
<br />
==Definition==<br />
<br />
The definition of the <code>Monoid</code> type class (defined in<br />
<code>Data.Monoid</code>; [http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Monoid.html haddock]) is:<br />
<br />
<haskell><br />
class Monoid a where<br />
mempty :: a<br />
mappend :: a -> a -> a<br />
<br />
mconcat :: [a] -> a<br />
mconcat = foldr mappend mempty<br />
</haskell><br />
<br />
The <code>mempty</code> value specifies the identity element of the monoid, and <code>mappend</code><br />
is the binary operation. The default definition for <code>mconcat</code><br />
“reduces” a list of elements by combining them all with <code>mappend</code>,<br />
using a right fold. It is only in the <code>Monoid</code> class so that specific<br />
instances have the option of providing an alternative, more efficient<br />
implementation; usually, you can safely ignore <code>mconcat</code> when creating<br />
a <code>Monoid</code> instance, since its default definition will work just fine.<br />
<br />
The <code>Monoid</code> methods are rather unfortunately named; they are inspired<br />
by the list instance of <code>Monoid</code>, where indeed <code>mempty = []</code> and <code>mappend = (++)</code>, but this is misleading since many<br />
monoids have little to do with appending (see these [http://thread.gmane.org/gmane.comp.lang.haskell.cafe/50590 Comments from OCaml Hacker Brian Hurt] on the haskell-cafe mailing list).<br />
<br />
==Laws==<br />
<br />
Of course, every <code>Monoid</code> instance should actually be a monoid in the<br />
mathematical sense, which implies these laws:<br />
<br />
<haskell><br />
mempty `mappend` x = x<br />
x `mappend` mempty = x<br />
(x `mappend` y) `mappend` z = x `mappend` (y `mappend` z)<br />
</haskell><br />
<br />
==Instances==<br />
<br />
There are quite a few interesting <code>Monoid</code> instances defined in <code>Data.Monoid</code>.<br />
<br />
<ul><br />
<li><code>[a]</code> is a <code>Monoid</code>, with <code>mempty = []</code> and <code>mappend = (++)</code>. It is not hard to check that <code>(x ++ y) ++ z = x ++ (y ++ z)</code> for any lists <code>x</code>, <code>y</code>, and <code>z</code>, and that the empty list is the identity: <code>[] ++ x = x ++ [] = x</code>.</li><br />
<br />
<li>As noted previously, we can make a monoid out of any numeric type under either addition or multiplication. However, since we can’t have two instances for the same type, <code>Data.Monoid</code> provides two <code>newtype</code> wrappers, <code>Sum</code> and <code>Product</code>, with appropriate <code>Monoid</code> instances.<br />
<br />
<haskell><br />
> getSum (mconcat . map Sum $ [1..5])<br />
15<br />
> getProduct (mconcat . map Product $ [1..5])<br />
120<br />
</haskell><br />
<br />
This example code is silly, of course; we could just write<br />
<code>sum [1..5]</code> and <code>product [1..5]</code>. Nevertheless, these instances are useful in more generalized settings, as we will see in the [[Foldable|section on <code>Foldable</code>]].</li><br />
<br />
<li><code>Any</code> and <code>All</code> are <code>newtype</code> wrappers providing <code>Monoid</code> instances for <code>Bool</code> (under disjunction and conjunction, respectively).</li><br />
<br />
<li> There are three instances for <code>Maybe</code>: a basic instance which lifts a <code>Monoid</code> instance for <code>a</code> to an instance for <code>Maybe a</code>, and two <code>newtype</code> wrappers <code>First</code> and <code>Last</code> for which <code>mappend</code> selects the first (respectively last) non-<code>Nothing</code> item.</li><br />
<br />
<li><code>Endo a</code> is a newtype wrapper for functions <code>a -> a</code>, which form a monoid under composition.</li><br />
<br />
<li>There are several ways to “lift” <code>Monoid</code> instances to instances with additional structure. We have already seen that an instance for <code>a</code> can be lifted to an instance for <code>Maybe a</code>. There are also tuple instances: if <code>a</code> and <code>b</code> are instances of <code>Monoid</code>, then so is <code>(a,b)</code>, using the monoid operations for <code>a</code> and <code>b</code> in the obvious pairwise manner. Finally, if <code>a</code> is a <code>Monoid</code>, then so is the function type <code>e -> a</code> for any <code>e</code>; in particular, <code>g `mappend` h</code> is the function which applies both <code>g</code> and <code>h</code> to its argument and then combines the results using the underlying <code>Monoid</code> instance for <code>a</code>. This can be quite useful and elegant (see [http://thread.gmane.org/gmane.comp.lang.haskell.cafe/52416 example]).</li><br />
<br />
<li>The type <code>Ordering = LT || EQ || GT</code> is a <code>Monoid</code>, defined in such a way that <code>mconcat (zipWith compare xs ys)</code> computes the lexicographic ordering of <code>xs</code> and <code>ys</code> (if <code>xs</code> and <code>ys</code> have the same length). In particular, <code>mempty = EQ</code>, and <code>mappend</code> evaluates to its leftmost non-<code>EQ</code> argument (or <code>EQ</code> if both arguments are <code>EQ</code>). This can be used together with the function instance of <code>Monoid</code> to do some clever things ([http://www.reddit.com/r/programming/comments/7cf4r/monoids_in_my_programming_language/c06adnx example]).</li><br />
<br />
<li>There are also <code>Monoid</code> instances for several standard data structures in the containers library ([http://hackage.haskell.org/packages/archive/containers/0.2.0.0/doc/html/index.html haddock]), including <code>Map</code>, <code>Set</code>, and <code>Sequence</code>.</li><br />
</ul><br />
<br />
<code>Monoid</code> is also used to enable several other type class instances.<br />
As noted previously, we can use <code>Monoid</code> to make <code>((,) e)</code> an instance of <code>Applicative</code>:<br />
<br />
<haskell><br />
instance Monoid e => Applicative ((,) e) where<br />
pure x = (mempty, x)<br />
(u, f) <*> (v, x) = (u `mappend` v, f x)<br />
</haskell><br />
<br />
<code>Monoid</code> can be similarly used to make <code>((,) e)</code> an instance of <code>Monad</code> as well; this is known as the ''writer monad''. As we’ve already seen, <code>Writer</code> and <code>WriterT</code> are a newtype wrapper and transformer for this monad, respectively.<br />
<br />
<code>Monoid</code> also plays a key role in the <code>Foldable</code> type class (see section [[#Foldable|Foldable]]).<br />
<br />
==Other monoidal classes: Alternative, MonadPlus, ArrowPlus==<br />
<br />
The <code>Alternative</code> type class ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Applicative.html#g:2 haddock])<br />
is for <code>Applicative</code> functors which also have<br />
a monoid structure:<br />
<br />
<haskell><br />
class Applicative f => Alternative f where<br />
empty :: f a<br />
(<|>) :: f a -> f a -> f a<br />
</haskell><br />
<br />
Of course, instances of <code>Alternative</code> should satisfy the monoid laws.<br />
<br />
Likewise, <code>MonadPlus</code> ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad.html#t:MonadPlus haddock])<br />
is for <code>Monad</code>s with a monoid structure:<br />
<br />
<haskell><br />
class Monad m => MonadPlus m where<br />
mzero :: m a<br />
mplus :: m a -> m a -> m a<br />
</haskell><br />
<br />
The <code>MonadPlus</code> documentation states that it is intended to model<br />
monads which also support “choice and failure”; in addition to the<br />
monoid laws, instances of <code>MonadPlus</code> are expected to satisfy<br />
<br />
<haskell><br />
mzero >>= f = mzero<br />
v >> mzero = mzero<br />
</haskell><br />
<br />
which explains the sense in which <code>mzero</code> denotes failure. Since<br />
<code>mzero</code> should be the identity for <code>mplus</code>, the computation <code>m1 `mplus` m2</code> succeeds (evaluates to something other than <code>mzero</code>) if<br />
either <code>m1</code> or <code>m2</code> does; so <code>mplus</code> represents choice. The <code>guard</code><br />
function can also be used with instances of <code>MonadPlus</code>; it requires a<br />
condition to be satisfied and fails (using <code>mzero</code>) if it is not. A<br />
simple example of a <code>MonadPlus</code> instance is <code>[]</code>, which is exactly the<br />
same as the <code>Monoid</code> instance for <code>[]</code>: the empty list represents<br />
failure, and list concatenation represents choice. In general,<br />
however, a <code>MonadPlus</code> instance for a type need not be the same as its<br />
<code>Monoid</code> instance; <code>Maybe</code> is an example of such a type. A great<br />
introduction to the <code>MonadPlus</code> type class, with interesting examples<br />
of its use, is Doug Auclair’s ''MonadPlus: What a Super Monad!'' in [http://www.haskell.org/wikiupload/6/6a/TMR-Issue11.pdf the Monad.Reader issue 11].<br />
<br />
There used to be a type class called <code>MonadZero</code> containing only<br />
<code>mzero</code>, representing monads with failure. The <code>do</code>-notation requires<br />
some notion of failure to deal with failing pattern matches.<br />
Unfortunately, <code>MonadZero</code> was scrapped in favor of adding the <code>fail</code><br />
method to the <code>Monad</code> class. If we are lucky, someday <code>MonadZero</code> will<br />
be restored, and <code>fail</code> will be banished to the bit bucket where it<br />
belongs (see [[MonadPlus reform proposal]]). The idea is that any<br />
<code>do</code>-block which uses pattern matching (and hence may fail) would require<br />
a <code>MonadZero</code> constraint; otherwise, only a <code>Monad</code> constraint would be<br />
required.<br />
<br />
Finally, <code>ArrowZero</code> and <code>ArrowPlus</code> ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html#t:ArrowZero haddock])<br />
represent <code>Arrow</code>s ([[#Arrow|see below]]) with a<br />
monoid structure:<br />
<br />
<haskell><br />
class Arrow arr => ArrowZero arr where<br />
zeroArrow :: b `arr` c<br />
<br />
class ArrowZero arr => ArrowPlus arr where<br />
(<+>) :: (b `arr` c) -> (b `arr` c) -> (b `arr` c)<br />
</haskell><br />
<br />
==Further reading==<br />
<br />
Monoids have gotten a fair bit of attention recently, ultimately due<br />
to<br />
[http://enfranchisedmind.com/blog/posts/random-thoughts-on-haskell/ a blog post by Brian Hurt], in which he<br />
complained about the fact that the names of many Haskell type classes<br />
(<code>Monoid</code> in particular) are taken from abstract mathematics. This<br />
resulted in [http://thread.gmane.org/gmane.comp.lang.haskell.cafe/50590 a long haskell-cafe thread]<br />
arguing the point and discussing monoids in general.<br />
<br />
{{note|May its name live forever.}}<br />
<br />
However, this was quickly followed by several blog posts about<br />
<code>Monoid</code> {{noteref}}. First, Dan Piponi<br />
wrote a great introductory post, [http://blog.sigfpe.com/2009/01/haskell-monoids-and-their-uses.html Haskell Monoids and their Uses]. This was quickly followed by<br />
Heinrich Apfelmus’s [http://apfelmus.nfshost.com/monoid-fingertree.html Monoids and Finger Trees], an accessible exposition of<br />
Hinze and Paterson’s [http://www.soi.city.ac.uk/%7Eross/papers/FingerTree.html classic paper on 2-3 finger trees], which makes very clever<br />
use of <code>Monoid</code> to implement an elegant and generic data structure.<br />
Dan Piponi then wrote two fascinating articles about using <code>Monoids</code><br />
(and finger trees): [http://blog.sigfpe.com/2009/01/fast-incremental-regular-expression.html Fast Incremental Regular Expressions] and [http://blog.sigfpe.com/2009/01/beyond-regular-expressions-more.html Beyond Regular Expressions]<br />
<br />
In a similar vein, David Place’s article on improving <code>Data.Map</code> in<br />
order to compute incremental folds (see [http://www.haskell.org/sitewiki/images/6/6a/TMR-Issue11.pdf the Monad Reader issue 11])<br />
is also a<br />
good example of using <code>Monoid</code> to generalize a data structure.<br />
<br />
Some other interesting examples of <code>Monoid</code> use include [http://www.reddit.com/r/programming/comments/7cf4r/monoids_in_my_programming_language/c06adnx building elegant list sorting combinators],<br />
[http://byorgey.wordpress.com/2008/04/17/collecting-unstructured-information-with-the-monoid-of-partial-knowledge/ collecting unstructured information],<br />
and a brilliant series of posts by Chung-Chieh Shan and Dylan Thurston<br />
using <code>Monoid</code>s to [http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers1/ elegantly solve a difficult combinatorial puzzle] (followed by<br />
[http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers2/ part 2],<br />
[http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers3/ part 3],<br />
[http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers4/ part 4]).<br />
<br />
As unlikely as it sounds, monads can actually be viewed as a sort of<br />
monoid, with <code>join</code> playing the role of the binary operation and<br />
<code>return</code> the role of the identity; see [http://blog.sigfpe.com/2008/11/from-monoids-to-monads.html Dan Piponi’s blog post].<br />
<br />
=Foldable=<br />
<br />
The <code>Foldable</code> class, defined in the <code>Data.Foldable</code><br />
module ([http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Foldable.html haddock]), abstracts over containers which can be<br />
“folded” into a summary value. This allows such folding operations<br />
to be written in a container-agnostic way.<br />
<br />
==Definition==<br />
<br />
The definition of the <code>Foldable</code> type class is:<br />
<br />
<haskell><br />
class Foldable t where<br />
fold :: Monoid m => t m -> m<br />
foldMap :: Monoid m => (a -> m) -> t a -> m<br />
<br />
foldr :: (a -> b -> b) -> b -> t a -> b<br />
foldl :: (a -> b -> a) -> a -> t b -> a<br />
foldr1 :: (a -> a -> a) -> t a -> a<br />
foldl1 :: (a -> a -> a) -> t a -> a<br />
</haskell><br />
<br />
This may look complicated, but in fact, to make a <code>Foldable</code> instance<br />
you only need to implement one method: your choice of <code>foldMap</code> or<br />
<code>foldr</code>. All the other methods have default implementations in terms<br />
of these, and are presumably included in the class in case more<br />
efficient implementations can be provided.<br />
<br />
==Instances and examples==<br />
<br />
The type of <code>foldMap</code> should make it clear what it is supposed to do:<br />
given a way to convert the data in a container into a <code>Monoid</code> (a<br />
function <code>a -> m</code>) and a container of <code>a</code>’s (<code>t a</code>), <code>foldMap</code><br />
provides a way to iterate over the entire contents of the container,<br />
converting all the <code>a</code>’s to <code>m</code>’s and combining all the <code>m</code>’s with<br />
<code>mappend</code>. The following code shows two examples: a simple<br />
implementation of <code>foldMap</code> for lists, and a binary tree example<br />
provided by the <code>Foldable</code> documentation.<br />
<br />
<haskell><br />
instance Foldable [] where<br />
foldMap g = mconcat . map g<br />
<br />
data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)<br />
<br />
instance Foldable Tree where<br />
foldMap f Empty = mempty<br />
foldMap f (Leaf x) = f x<br />
foldMap f (Node l k r) = foldMap f l `mappend` f k `mappend` foldMap f r<br />
</haskell><br />
<br />
The <code>foldr</code> function has a type similar to the <code>foldr</code> found in the <code>Prelude</code>, but<br />
more general, since the <code>foldr</code> in the <code>Prelude</code> works only on lists.<br />
<br />
The <code>Foldable</code> module also provides instances for <code>Maybe</code> and <code>Array</code>;<br />
additionally, many of the data structures found in the standard [http://hackage.haskell.org/package/containers containers library] (for example, <code>Map</code>, <code>Set</code>, <code>Tree</code>,<br />
and <code>Sequence</code>) provide their own <code>Foldable</code> instances.<br />
<br />
==Derived folds==<br />
<br />
Given an instance of <code>Foldable</code>, we can write generic,<br />
container-agnostic functions such as:<br />
<br />
<haskell><br />
-- Compute the size of any container.<br />
containerSize :: Foldable f => f a -> Int<br />
containerSize = getSum . foldMap (const (Sum 1))<br />
<br />
-- Compute a list of elements of a container satisfying a predicate.<br />
filterF :: Foldable f => (a -> Bool) -> f a -> [a]<br />
filterF p = foldMap (\a -> if p a then [a] else [])<br />
<br />
-- Get a list of all the Strings in a container which include the<br />
-- letter a.<br />
aStrings :: Foldable f => f String -> [String]<br />
aStrings = filterF (elem 'a')<br />
</haskell><br />
<br />
The <code>Foldable</code> module also provides a large number of predefined<br />
folds, many of which are generalized versions of <code>Prelude</code> functions of the<br />
same name that only work on lists: <code>concat</code>, <code>concatMap</code>, <code>and</code>,<br />
<code>or</code>, <code>any</code>, <code>all</code>, <code>sum</code>, <code>product</code>, <code>maximum</code>(<code>By</code>),<br />
<code>minimum</code>(<code>By</code>), <code>elem</code>, <code>notElem</code>, and <code>find</code>.<br />
<br />
The important function <code>toList</code> is also provided, which turns any <code>Foldable</code> structure into a list of its elements in left-right order; it works by folding with the list monoid.<br />
<br />
There are also generic functions that work with <code>Applicative</code> or<br />
<code>Monad</code> instances to generate some sort of computation from each<br />
element in a container, and then perform all the side effects from<br />
those computations, discarding the results: <code>traverse_</code>, <code>sequenceA_</code>,<br />
and others. The results must be discarded because the <code>Foldable</code><br />
class is too weak to specify what to do with them: we cannot, in<br />
general, make an arbitrary <code>Applicative</code> or <code>Monad</code> instance into a <code>Monoid</code>, but we can make <code>m ()</code> into a <code>Monoid</code> for any such <code>m</code>. If we do have an <code>Applicative</code> or <code>Monad</code> with a monoid<br />
structure—that is, an <code>Alternative</code> or a <code>MonadPlus</code>—then we can<br />
use the <code>asum</code> or <code>msum</code> functions, which can combine the results as<br />
well. Consult the [http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Foldable.html <code>Foldable</code> documentation] for<br />
more details on any of these functions.<br />
<br />
Note that the <code>Foldable</code> operations always forget the structure of<br />
the container being folded. If we start with a container of type <code>t a</code> for some <code>Foldable t</code>, then <code>t</code> will never appear in the output<br />
type of any operations defined in the <code>Foldable</code> module. Many times<br />
this is exactly what we want, but sometimes we would like to be able<br />
to generically traverse a container while preserving its<br />
structure—and this is exactly what the <code>Traversable</code> class provides,<br />
which will be discussed in the next section.<br />
<br />
{{Exercises|<br />
# Implement <code>toList :: Foldable f {{=}}> f a -> [a]</code>.<br />
# Pick some of the following functions to implement: <code>concat</code>, <code>concatMap</code>, <code>and</code>, <code>or</code>, <code>any</code>, <code>all</code>, <code>sum</code>, <code>product</code>, <code>maximum</code>(<code>By</code>), <code>minimum</code>(<code>By</code>), <code>elem</code>, <code>notElem</code>, and <code>find</code>. Figure out how they generalize to <code>Foldable</code> and come up with elegant implementations using <code>fold</code> or <code>foldMap</code> along with appropriate <code>Monoid</code> instances.<br />
}}<br />
<br />
==Foldable actually isn't==<br />
<br />
The generic term "fold" is often used to refer to the more technical concept of [[Catamorphisms|catamorphism]]. Intuitively, given a way to summarize "one level of structure" (where recursive subterms have already been replaced with their summaries), a catamorphism can summarize an entire recursive structure. It is important to realize that <code>Foldable</code> does <i>not</i> correspond to catamorphisms, but to something weaker. In particular, <code>Foldable</code> allows observing only the left-right order of elements within a structure, not the actual structure itself. Put another way, every use of <code>Foldable</code> can be expressed in terms of <code>toList</code>. For example, <code>fold</code> itself is equivalent to <code>mconcat . toList</code>.<br />
<br />
This is sufficient for many tasks, but not all. For example, consider trying to compute the depth of a <code>Tree</code>: try as we might, there is no way to implement it using <code>Foldable</code>. However, it <i>can</i> be implemented as a catamorphism.<br />
<br />
==Further reading==<br />
<br />
The <code>Foldable</code> class had its genesis in [http://www.soi.city.ac.uk/~ross/papers/Applicative.html McBride and Paterson’s paper]<br />
introducing <code>Applicative</code>, although it has<br />
been fleshed out quite a bit from the form in the paper.<br />
<br />
An interesting use of <code>Foldable</code> (as well as <code>Traversable</code>) can be<br />
found in Janis Voigtländer’s paper [http://doi.acm.org/10.1145/1480881.1480904 Bidirectionalization for free!].<br />
<br />
=Traversable=<br />
<br />
==Definition==<br />
<br />
The <code>Traversable</code> type class, defined in the <code>Data.Traversable</code><br />
module ([http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Traversable.html haddock]), is:<br />
<br />
<haskell><br />
class (Functor t, Foldable t) => Traversable t where<br />
traverse :: Applicative f => (a -> f b) -> t a -> f (t b)<br />
sequenceA :: Applicative f => t (f a) -> f (t a)<br />
mapM :: Monad m => (a -> m b) -> t a -> m (t b)<br />
sequence :: Monad m => t (m a) -> m (t a)<br />
</haskell><br />
<br />
As you can see, every <code>Traversable</code> is also a foldable functor. Like<br />
<code>Foldable</code>, there is a lot in this type class, but making instances is<br />
actually rather easy: one need only implement <code>traverse</code> or<br />
<code>sequenceA</code>; the other methods all have default implementations in<br />
terms of these functions. A good exercise is to figure out what the default<br />
implementations should be: given either <code>traverse</code> or <code>sequenceA</code>, how<br />
would you define the other three methods? (Hint for <code>mapM</code>:<br />
<code>Control.Applicative</code> exports the <code>WrapMonad</code> newtype, which makes any<br />
<code>Monad</code> into an <code>Applicative</code>. The <code>sequence</code> function can be implemented in terms<br />
of <code>mapM</code>.)<br />
<br />
==Intuition==<br />
<br />
The key method of the <code>Traversable</code> class, and the source of its<br />
unique power, is <code>sequenceA</code>. Consider its type:<br />
<haskell><br />
sequenceA :: Applicative f => t (f a) -> f (t a)<br />
</haskell><br />
This answers the fundamental question: when can we commute two<br />
functors? For example, can we turn a tree of lists into a list of<br />
trees? (Answer: yes, in two ways. Figuring out what they are, and<br />
why, is left as an exercise. A much more challenging question is<br />
whether a list of trees can be turned into a tree of lists.)<br />
<br />
The ability to compose two monads depends crucially on this ability to<br />
commute functors. Intuitively, if we want to build a composed monad<br />
<code>M a = m (n a)</code> out of monads <code>m</code> and <code>n</code>, then to be able to<br />
implement <code>join :: M (M a) -> M a</code>, that is,<br />
<code>join :: m (n (m (n a))) -> m (n a)</code>, we have to be able to commute<br />
the <code>n</code> past the <code>m</code> to get <code>m (m (n (n a)))</code>, and then we can use the<br />
<code>join</code>s for <code>m</code> and <code>n</code> to produce something of type <code>m (n a)</code>. See<br />
[http://web.cecs.pdx.edu/~mpj/pubs/springschool.html Mark Jones’s paper] for more details.<br />
<br />
==Instances and examples==<br />
<br />
What’s an example of a <code>Traversable</code> instance?<br />
The following code shows an example instance for the same<br />
<code>Tree</code> type used as an example in the previous <code>Foldable</code> section. It<br />
is instructive to compare this instance with a <code>Functor</code> instance for<br />
<code>Tree</code>, which is also shown.<br />
<br />
<haskell><br />
data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)<br />
<br />
instance Traversable Tree where<br />
traverse g Empty = pure Empty<br />
traverse g (Leaf x) = Leaf <$> g x<br />
traverse g (Node l x r) = Node <$> traverse g l<br />
<*> g x<br />
<*> traverse g r<br />
<br />
instance Functor Tree where<br />
fmap g Empty = Empty<br />
fmap g (Leaf x) = Leaf $ g x<br />
fmap g (Node l x r) = Node (fmap g l)<br />
(g x)<br />
(fmap g r)<br />
</haskell><br />
<br />
It should be clear that the <code>Traversable</code> and <code>Functor</code> instances for<br />
<code>Tree</code> are almost identical; the only difference is that the <code>Functor</code><br />
instance involves normal function application, whereas the<br />
applications in the <code>Traversable</code> instance take place within an<br />
<code>Applicative</code> context, using <code>(<$>)</code> and <code>(<*>)</code>. In fact, this will<br />
be<br />
true for any type.<br />
<br />
Any <code>Traversable</code> functor is also <code>Foldable</code>, and a <code>Functor</code>. We can see<br />
this not only from the class declaration, but by the fact that we can<br />
implement the methods of both classes given only the <code>Traversable</code><br />
methods. A good exercise is to implement <code>fmap</code> and <code>foldMap</code> using<br />
only the <code>Traversable</code> methods; the implementations are surprisingly<br />
elegant. The <code>Traversable</code> module provides these<br />
implementations as <code>fmapDefault</code> and <code>foldMapDefault</code>.<br />
<br />
The standard libraries provide a number of <code>Traversable</code> instances,<br />
including instances for <code>[]</code>, <code>Maybe</code>, <code>Map</code>, <code>Tree</code>, and <code>Sequence</code>.<br />
Notably, <code>Set</code> is not <code>Traversable</code>, although it is <code>Foldable</code>.<br />
<br />
==Further reading==<br />
<br />
The <code>Traversable</code> class also had its genesis in [http://www.soi.city.ac.uk/~ross/papers/Applicative.html McBride and Paterson’s <code>Applicative</code> paper],<br />
and is described in more detail in Gibbons and Oliveira, [http://www.comlab.ox.ac.uk/jeremy.gibbons/publications/iterator.pdf The Essence of the Iterator Pattern],<br />
which also contains a wealth of references to related work.<br />
<br />
=Category=<br />
<br />
<code>Category</code> is a relatively recent addition to the Haskell standard libraries. It generalizes the notion of function composition to general “morphisms”.<br />
<br />
{{note|GHC 7.6.1 changed its rules regarding types and type variables. Now, any operator at the type level is treated as a type ''constructor'' rather than a type ''variable''; prior to GHC 7.6.1 it was possible to use <code>(~&gt;)</code> instead of <code>`arr`</code>. For more information, see [http://thread.gmane.org/gmane.comp.lang.haskell.glasgow.user/21350 the discussion on the GHC-users mailing list]. For a new approach to nice arrow notation that works with GHC 7.6.1, see [http://article.gmane.org/gmane.comp.lang.haskell.glasgow.user/22615 this messsage] and also [http://article.gmane.org/gmane.comp.lang.haskell.glasgow.user/22616 this message] from Edward Kmett, though for simplicity I haven't adopted it here.}}<br />
The definition of the <code>Category</code> type class (from<br />
<code>Control.Category</code>—[http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Category.html haddock]) is shown below. For ease of reading, note that I have used an infix type variable <code>`arr`</code>, in parallel with the infix function type constructor <code>(->)</code>. {{noteref}} This syntax is not part of Haskell 2010. The second definition shown is the one used in the standard libraries. For the remainder of this document, I will use the infix type constructor <code>`arr`</code> for <code>Category</code> as well as <code>Arrow</code>.<br />
<br />
<haskell><br />
class Category arr where<br />
id :: a `arr` a<br />
(.) :: (b `arr` c) -> (a `arr` b) -> (a `arr` c)<br />
<br />
-- The same thing, with a normal (prefix) type constructor<br />
class Category cat where<br />
id :: cat a a<br />
(.) :: cat b c -> cat a b -> cat a c<br />
</haskell><br />
<br />
Note that an instance of <code>Category</code> should be a type constructor which takes two type arguments, that is, something of kind <code>* -> * -> *</code>. It is instructive to imagine the type constructor variable <code>cat</code> replaced by the function constructor <code>(->)</code>: indeed, in this case we recover precisely the familiar identity function <code>id</code> and function composition operator <code>(.)</code> defined in the standard <code>Prelude</code>.<br />
<br />
Of course, the <code>Category</code> module provides exactly such an instance of<br />
<code>Category</code> for <code>(->)</code>. But it also provides one other instance, shown below, which should be familiar from the previous discussion of the <code>Monad</code> laws. <code>Kleisli m a b</code>, as defined in the <code>Control.Arrow</code> module, is just a <code>newtype</code> wrapper around <code>a -> m b</code>.<br />
<br />
<haskell><br />
newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b }<br />
<br />
instance Monad m => Category (Kleisli m) where<br />
id = Kleisli return<br />
Kleisli g . Kleisli h = Kleisli (h >=> g)<br />
</haskell><br />
<br />
The only law that <code>Category</code> instances should satisfy is that <code>id</code> and <code>(.)</code> should form a monoid—that is, <code>id</code> should be the identity of <code>(.)</code>, and <code>(.)</code> should be associative.<br />
<br />
Finally, the <code>Category</code> module exports two additional operators:<br />
<code>(<<<)</code>, which is just a synonym for <code>(.)</code>, and <code>(>>>)</code>, which is <code>(.)</code> with its arguments reversed. (In previous versions of the libraries, these operators were defined as part of the <code>Arrow</code> class.)<br />
<br />
==Further reading==<br />
<br />
The name <code>Category</code> is a bit misleading, since the <code>Category</code> class cannot represent arbitrary categories, but only categories whose objects are objects of <code>Hask</code>, the category of Haskell types. For a more general treatment of categories within Haskell, see the [http://hackage.haskell.org/package/category-extras category-extras package]. For more about category theory in general, see the excellent [http://en.wikibooks.org/wiki/Haskell/Category_theory Haskell wikibook page],<br />
[http://books.google.com/books/about/Category_theory.html?id=-MCJ6x2lC7oC Steve Awodey’s new book], Benjamin Pierce’s [http://books.google.com/books/about/Basic_category_theory_for_computer_scien.html?id=ezdeaHfpYPwC Basic category theory for computer scientists], or [http://folli.loria.fr/cds/1999/esslli99/courses/barr-wells.html Barr and Wells’s category theory lecture notes]. [http://dekudekuplex.wordpress.com/2009/01/19/motivating-learning-category-theory-for-non-mathematicians/ Benjamin Russell’s blog post]<br />
is another good source of motivation and category theory links. You certainly don’t need to know any category theory to be a successful and productive Haskell programmer, but it does lend itself to much deeper appreciation of Haskell’s underlying theory.<br />
<br />
=Arrow=<br />
<br />
The <code>Arrow</code> class represents another abstraction of computation, in a<br />
similar vein to <code>Monad</code> and <code>Applicative</code>. However, unlike <code>Monad</code><br />
and <code>Applicative</code>, whose types only reflect their output, the type of<br />
an <code>Arrow</code> computation reflects both its input and output. Arrows<br />
generalize functions: if <code>arr</code> is an instance of <code>Arrow</code>, a value of<br />
type <code>b `arr` c</code> can be thought of as a computation which takes values of<br />
type <code>b</code> as input, and produces values of type <code>c</code> as output. In the<br />
<code>(->)</code> instance of <code>Arrow</code> this is just a pure function; in general, however,<br />
an arrow may represent some sort of “effectful” computation.<br />
<br />
==Definition==<br />
<br />
The definition of the <code>Arrow</code> type class, from<br />
<code>Control.Arrow</code> ([http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Arrow.html haddock]), is:<br />
<br />
<haskell><br />
class Category arr => Arrow arr where<br />
arr :: (b -> c) -> (b `arr` c)<br />
first :: (b `arr` c) -> ((b, d) `arr` (c, d))<br />
second :: (b `arr` c) -> ((d, b) `arr` (d, c))<br />
(***) :: (b `arr` c) -> (b' `arr` c') -> ((b, b') `arr` (c, c'))<br />
(&&&) :: (b `arr` c) -> (b `arr` c') -> (b `arr` (c, c'))<br />
</haskell><br />
<br />
{{note|In versions of the <code>base</code><br />
package prior to version 4, there is no <code>Category</code> class, and the<br />
<code>Arrow</code> class includes the arrow composition operator <code>(>>>)</code>. It<br />
also includes <code>pure</code> as a synonym for <code>arr</code>, but this was removed<br />
since it conflicts with the <code>pure</code> from <code>Applicative</code>.}}<br />
<br />
The first thing to note is the <code>Category</code> class constraint, which<br />
means that we get identity arrows and arrow composition for free:<br />
given two arrows <code>g :: b `arr` c</code> and <code>h :: c `arr` d</code>, we can form their<br />
composition <code>g >>> h :: b `arr` d</code> {{noteref}}.<br />
<br />
As should be a familiar pattern by now, the only methods which must be<br />
defined when writing a new instance of <code>Arrow</code> are <code>arr</code> and <code>first</code>;<br />
the other methods have default definitions in terms of these, but are<br />
included in the <code>Arrow</code> class so that they can be overridden with more<br />
efficient implementations if desired.<br />
<br />
==Intuition==<br />
<br />
Let’s look at each of the arrow methods in turn. [http://www.haskell.org/arrows/ Ross Paterson’s web page on arrows] has nice diagrams which can help<br />
build intuition.<br />
<br />
* The <code>arr</code> function takes any function <code>b -> c</code> and turns it into a generalized arrow <code>b `arr` c</code>. The <code>arr</code> method justifies the claim that arrows generalize functions, since it says that we can treat any function as an arrow. It is intended that the arrow <code>arr g</code> is “pure” in the sense that it only computes <code>g</code> and has no “effects” (whatever that might mean for any particular arrow type).<br />
<br />
* The <code>first</code> method turns any arrow from <code>b</code> to <code>c</code> into an arrow from <code>(b,d)</code> to <code>(c,d)</code>. The idea is that <code>first g</code> uses <code>g</code> to process the first element of a tuple, and lets the second element pass through unchanged. For the function instance of <code>Arrow</code>, of course, <code>first g (x,y) = (g x, y)</code>.<br />
<br />
* The <code>second</code> function is similar to <code>first</code>, but with the elements of the tuples swapped. Indeed, it can be defined in terms of <code>first</code> using an auxiliary function <code>swap</code>, defined by <code>swap (x,y) = (y,x)</code>.<br />
<br />
* The <code>(***)</code> operator is “parallel composition” of arrows: it takes two arrows and makes them into one arrow on tuples, which has the behavior of the first arrow on the first element of a tuple, and the behavior of the second arrow on the second element. The mnemonic is that <code>g *** h</code> is the ''product'' (hence <code>*</code>) of <code>g</code> and <code>h</code>. For the function instance of <code>Arrow</code>, we define <code>(g *** h) (x,y) = (g x, h y)</code>. The default implementation of <code>(***)</code> is in terms of <code>first</code>, <code>second</code>, and sequential arrow composition <code>(>>>)</code>. The reader may also wish to think about how to implement <code>first</code> and <code>second</code> in terms of <code>(***)</code>.<br />
<br />
* The <code>(&&&)</code> operator is “fanout composition” of arrows: it takes two arrows <code>g</code> and <code>h</code> and makes them into a new arrow <code>g &&& h</code> which supplies its input as the input to both <code>g</code> and <code>h</code>, returning their results as a tuple. The mnemonic is that <code>g &&& h</code> performs both <code>g</code> ''and'' <code>h</code> (hence <code>&</code>) on its input. For functions, we define <code>(g &&& h) x = (g x, h x)</code>.<br />
<br />
==Instances==<br />
<br />
The <code>Arrow</code> library itself only provides two <code>Arrow</code> instances, both<br />
of which we have already seen: <code>(->)</code>, the normal function<br />
constructor, and <code>Kleisli m</code>, which makes functions of<br />
type <code>a -> m b</code> into <code>Arrow</code>s for any <code>Monad m</code>. These instances are:<br />
<br />
<haskell><br />
instance Arrow (->) where<br />
arr g = g<br />
first g (x,y) = (g x, y)<br />
<br />
newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b }<br />
<br />
instance Monad m => Arrow (Kleisli m) where<br />
arr f = Kleisli (return . f)<br />
first (Kleisli f) = Kleisli (\ ~(b,d) -> do c <- f b<br />
return (c,d) )<br />
</haskell><br />
<br />
==Laws==<br />
<br />
{{note|See [http://dx.doi.org/10.1016/S0167-6423(99)00023-4 John Hughes: Generalising monads to arrows]; [http://homepages.inf.ed.ac.uk/wadler/papers/arrows/arrows.pdf Sam Lindley, Philip Wadler, Jeremy Yallop: The arrow calculus]; [http://www.soi.city.ac.uk/~ross/papers/fop.html Ross Paterson: Programming with Arrows].}}<br />
<br />
There are quite a few laws that instances of <code>Arrow</code> should<br />
satisfy {{noteref}}:<br />
<br />
<haskell><br />
arr id = id<br />
arr (h . g) = arr g >>> arr h<br />
first (arr g) = arr (g *** id)<br />
first (g >>> h) = first g >>> first h<br />
first g >>> arr (id *** h) = arr (id *** h) >>> first g<br />
first g >>> arr fst = arr fst >>> g<br />
first (first g) >>> arr assoc = arr assoc >>> first g<br />
<br />
assoc ((x,y),z) = (x,(y,z))<br />
</haskell><br />
<br />
Note that this version of the laws is slightly different than the laws given in the<br />
first two above references, since several of the laws have now been<br />
subsumed by the <code>Category</code> laws (in particular, the requirements that<br />
<code>id</code> is the identity arrow and that <code>(>>>)</code> is associative). The laws<br />
shown here follow those in Paterson’s Programming with Arrows, which uses the<br />
<code>Category</code> class.<br />
<br />
{{note|Unless category-theory-induced insomnolence is your cup of tea.}}<br />
<br />
The reader is advised not to lose too much sleep over the <code>Arrow</code><br />
laws {{noteref}}, since it is not essential to understand them in order to<br />
program with arrows. There are also laws that <code>ArrowChoice</code>,<br />
<code>ArrowApply</code>, and <code>ArrowLoop</code> instances should satisfy; the interested<br />
reader should consult [http://www.soi.city.ac.uk/~ross/papers/fop.html Paterson: Programming with Arrows].<br />
<br />
==ArrowChoice==<br />
<br />
Computations built using the <code>Arrow</code> class, like those built using<br />
the <code>Applicative</code> class, are rather inflexible: the structure of the computation<br />
is fixed at the outset, and there is no ability to choose between<br />
alternate execution paths based on intermediate results.<br />
The <code>ArrowChoice</code> class provides exactly such an ability:<br />
<br />
<haskell><br />
class Arrow arr => ArrowChoice arr where<br />
left :: (b `arr` c) -> (Either b d `arr` Either c d)<br />
right :: (b `arr` c) -> (Either d b `arr` Either d c)<br />
(+++) :: (b `arr` c) -> (b' `arr` c') -> (Either b b' `arr` Either c c')<br />
(|||) :: (b `arr` d) -> (c `arr` d) -> (Either b c `arr` d)<br />
</haskell><br />
<br />
A comparison of <code>ArrowChoice</code> to <code>Arrow</code> will reveal a striking<br />
parallel between <code>left</code>, <code>right</code>, <code>(+++)</code>, <code>(|||)</code> and <code>first</code>,<br />
<code>second</code>, <code>(***)</code>, <code>(&&&)</code>, respectively. Indeed, they are dual:<br />
<code>first</code>, <code>second</code>, <code>(***)</code>, and <code>(&&&)</code> all operate on product types<br />
(tuples), and <code>left</code>, <code>right</code>, <code>(+++)</code>, and <code>(|||)</code> are the<br />
corresponding operations on sum types. In general, these operations<br />
create arrows whose inputs are tagged with <code>Left</code> or <code>Right</code>, and can<br />
choose how to act based on these tags.<br />
<br />
* If <code>g</code> is an arrow from <code>b</code> to <code>c</code>, then <code>left g</code> is an arrow from <code>Either b d</code> to <code>Either c d</code>. On inputs tagged with <code>Left</code>, the <code>left g</code> arrow has the behavior of <code>g</code>; on inputs tagged with <code>Right</code>, it behaves as the identity.<br />
<br />
* The <code>right</code> function, of course, is the mirror image of <code>left</code>. The arrow <code>right g</code> has the behavior of <code>g</code> on inputs tagged with <code>Right</code>.<br />
<br />
* The <code>(+++)</code> operator performs “multiplexing”: <code>g +++ h</code> behaves as <code>g</code> on inputs tagged with <code>Left</code>, and as <code>h</code> on inputs tagged with <code>Right</code>. The tags are preserved. The <code>(+++)</code> operator is the ''sum'' (hence <code>+</code>) of two arrows, just as <code>(***)</code> is the product.<br />
<br />
* The <code>(|||)</code> operator is “merge” or “fanin”: the arrow <code>g ||| h</code> behaves as <code>g</code> on inputs tagged with <code>Left</code>, and <code>h</code> on inputs tagged with <code>Right</code>, but the tags are discarded (hence, <code>g</code> and <code>h</code> must have the same output type). The mnemonic is that <code>g ||| h</code> performs either <code>g</code> ''or'' <code>h</code> on its input.<br />
<br />
The <code>ArrowChoice</code> class allows computations to choose among a finite number of execution paths, based on intermediate results. The possible<br />
execution paths must be known in advance, and explicitly assembled with <code>(+++)</code> or <code>(|||)</code>. However, sometimes more flexibility is<br />
needed: we would like to be able to ''compute'' an arrow from intermediate results, and use this computed arrow to continue the computation. This is the power given to us by <code>ArrowApply</code>.<br />
<br />
==ArrowApply==<br />
<br />
The <code>ArrowApply</code> type class is:<br />
<br />
<haskell><br />
class Arrow arr => ArrowApply arr where<br />
app :: (b `arr` c, b) `arr` c<br />
</haskell><br />
<br />
If we have computed an arrow as the output of some previous<br />
computation, then <code>app</code> allows us to apply that arrow to an input,<br />
producing its output as the output of <code>app</code>. As an exercise, the<br />
reader may wish to use <code>app</code> to implement an alternative “curried”<br />
version, <code>app2 :: b `arr` ((b `arr` c) `arr` c)</code>.<br />
<br />
This notion of being able to ''compute'' a new computation<br />
may sound familiar:<br />
this is exactly what the monadic bind operator <code>(>>=)</code> does. It<br />
should not particularly come as a surprise that <code>ArrowApply</code> and<br />
<code>Monad</code> are exactly equivalent in expressive power. In particular,<br />
<code>Kleisli m</code> can be made an instance of <code>ArrowApply</code>, and any instance<br />
of <code>ArrowApply</code> can be made a <code>Monad</code> (via the <code>newtype</code> wrapper<br />
<code>ArrowMonad</code>). As an exercise, the reader may wish to try<br />
implementing these instances:<br />
<br />
<haskell><br />
instance Monad m => ArrowApply (Kleisli m) where<br />
app = -- exercise<br />
<br />
newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)<br />
<br />
instance ArrowApply a => Monad (ArrowMonad a) where<br />
return = -- exercise<br />
(ArrowMonad a) >>= k = -- exercise<br />
</haskell><br />
<br />
==ArrowLoop==<br />
<br />
The <code>ArrowLoop</code> type class is:<br />
<br />
<haskell><br />
class Arrow a => ArrowLoop a where<br />
loop :: a (b, d) (c, d) -> a b c<br />
<br />
trace :: ((b,d) -> (c,d)) -> b -> c<br />
trace f b = let (c,d) = f (b,d) in c<br />
</haskell><br />
<br />
It describes arrows that can use recursion to compute results, and is<br />
used to desugar the <code>rec</code> construct in arrow notation (described<br />
below).<br />
<br />
Taken by itself, the type of the <code>loop</code> method does not seem to tell<br />
us much. Its intention, however, is a generalization of the <code>trace</code><br />
function which is also shown. The <code>d</code> component of the first arrow’s<br />
output is fed back in as its own input. In other words, the arrow<br />
<code>loop g</code> is obtained by recursively “fixing” the second component of<br />
the input to <code>g</code>.<br />
<br />
It can be a bit difficult to grok what the <code>trace</code> function is doing.<br />
How can <code>d</code> appear on the left and right sides of the <code>let</code>? Well,<br />
this is Haskell’s laziness at work. There is not space here for a<br />
full explanation; the interested reader is encouraged to study the<br />
standard <code>fix</code> function, and to read [http://www.soi.city.ac.uk/~ross/papers/fop.html Paterson’s arrow tutorial].<br />
<br />
==Arrow notation==<br />
<br />
Programming directly with the arrow combinators can be painful,<br />
especially when writing complex computations which need to retain<br />
simultaneous reference to a number of intermediate results. With<br />
nothing but the arrow combinators, such intermediate results must be<br />
kept in nested tuples, and it is up to the programmer to remember<br />
which intermediate results are in which components, and to swap,<br />
reassociate, and generally mangle tuples as necessary. This problem<br />
is solved by the special arrow notation supported by GHC, similar to<br />
<code>do</code> notation for monads, that allows names to be assigned to<br />
intermediate results while building up arrow computations. An example<br />
arrow implemented using arrow notation, taken from<br />
Paterson, is:<br />
<br />
<haskell><br />
class ArrowLoop arr => ArrowCircuit arr where<br />
delay :: b -> (b `arr` b)<br />
<br />
counter :: ArrowCircuit arr => Bool `arr` Int<br />
counter = proc reset -> do<br />
rec output <- idA -< if reset then 0 else next<br />
next <- delay 0 -< output + 1<br />
idA -< output<br />
</haskell><br />
<br />
This arrow is intended to<br />
represent a recursively defined counter circuit with a reset line.<br />
<br />
There is not space here for a full explanation of arrow notation; the<br />
interested reader should consult<br />
[http://www.soi.city.ac.uk/~ross/papers/notation.html Paterson’s paper introducing the notation], or his later [http://www.soi.city.ac.uk/~ross/papers/fop.html tutorial which presents a simplified version].<br />
<br />
==Further reading==<br />
<br />
An excellent starting place for the student of arrows is the [http://www.haskell.org/arrows/ arrows web page], which contains an<br />
introduction and many references. Some key papers on arrows include<br />
Hughes’s original paper introducing arrows, [http://dx.doi.org/10.1016/S0167-6423(99)00023-4 Generalising monads to arrows], and [http://www.soi.city.ac.uk/~ross/papers/notation.html Paterson’s paper on arrow notation].<br />
<br />
Both Hughes and Paterson later wrote accessible tutorials intended for a broader<br />
audience: [http://www.soi.city.ac.uk/~ross/papers/fop.html Paterson: Programming with Arrows] and [http://www.cse.chalmers.se/~rjmh/afp-arrows.pdf Hughes: Programming with Arrows].<br />
<br />
Although Hughes’s goal in defining the <code>Arrow</code> class was to<br />
generalize <code>Monad</code>s, and it has been said that <code>Arrow</code> lies “between<br />
<code>Applicative</code> and <code>Monad</code>” in power, they are not directly<br />
comparable. The precise relationship remained in some confusion until<br />
[http://homepages.inf.ed.ac.uk/wadler/papers/arrows-and-idioms/arrows-and-idioms.pdf analyzed by Lindley, Wadler, and Yallop], who<br />
also invented a new calculus of arrows, based on the lambda calculus,<br />
which considerably simplifies the presentation of the arrow laws<br />
(see [http://homepages.inf.ed.ac.uk/wadler/papers/arrows/arrows.pdf The arrow calculus]).<br />
<br />
Some examples of <code>Arrow</code>s include [http://www.haskell.org/yampa/ Yampa], the<br />
[http://www.fh-wedel.de/~si/HXmlToolbox/ Haskell XML Toolkit], and the functional GUI library [[Grapefruit]].<br />
<br />
Some extensions to arrows have been explored; for example, the<br />
[http://www.cs.ru.nl/A.vanWeelden/bi-arrows/ <code>BiArrow</code>s of Alimarine et al.], for two-way instead of one-way<br />
computation.<br />
<br />
The Haskell wiki has [[Research papers/Monads and Arrows|links to many additional research papers relating to <code>Arrow</code>s]].<br />
<br />
=Comonad=<br />
<br />
The final type class we will examine is <code>Comonad</code>. The <code>Comonad</code> class<br />
is the categorical dual of <code>Monad</code>; that is, <code>Comonad</code> is like <code>Monad</code><br />
but with all the function arrows flipped. It is not actually in the<br />
standard Haskell libraries, but it has seen some interesting uses<br />
recently, so we include it here for completeness.<br />
<br />
==Definition==<br />
<br />
The <code>Comonad</code> type class, defined in the <code>Control.Comonad</code> module of<br />
the [http://hackage.haskell.org/package/comonad comonad library], is:<br />
<br />
<haskell><br />
class Functor w => Comonad w where<br />
extract :: f a -> a<br />
<br />
duplicate :: w a -> w (w a)<br />
duplicate = extend id<br />
<br />
extend :: (w a -> b) -> w a -> w b<br />
extend f = fmap f . duplicate<br />
</haskell><br />
<br />
As you can see, <code>extract</code> is the dual of <code>return</code>, <code>duplicate</code> is the dual of <code>join</code>, and <code>extend</code> is the dual of <code>(=<<)</code>. The definition of <code>Comonad</code> is a bit redundant, giving the programmer the choice on whether extend or duplicate are implemented; the other operation then has a default implementation.<br />
<br />
A prototypical example of a <code>Comonad</code> instance is:<br />
<br />
<haskell><br />
-- Infinite lazy streams<br />
data Stream a = Cons a (Stream a)<br />
<br />
-- 'duplicate' is like the list function 'tails'<br />
-- 'extend' computes a new Stream from an old, where the element<br />
-- at position n is computed as a function of everything from<br />
-- position n onwards in the old Stream<br />
instance Comonad Stream where<br />
extract (Cons x _) = x<br />
duplicate s@(Cons x xs) = Cons s (duplicate xs)<br />
extend g s@(Cons x xs) = Cons (g s) (extend g xs)<br />
-- = fmap g (duplicate s)<br />
</haskell><br />
<br />
==Further reading==<br />
<br />
Dan Piponi explains in a blog post what [http://blog.sigfpe.com/2006/12/evaluating-cellular-automata-is.html cellular automata have to do with comonads]. In another blog post, Conal Elliott has examined [http://conal.net/blog/posts/functional-interactive-behavior/ a comonadic formulation of functional reactive programming]. Sterling Clover’s blog post [http://fmapfixreturn.wordpress.com/2008/07/09/comonads-in-everyday-life/ Comonads in everyday life] explains the relationship between comonads and zippers, and how comonads can be used to design a menu system for a web site.<br />
<br />
Uustalu and Vene have a number of papers exploring ideas related to comonads and functional programming:<br />
* [http://dx.doi.org/10.1016/j.entcs.2008.05.029 Comonadic Notions of Computation]<br />
* [http://www.ioc.ee/~tarmo/papers/sfp01-book.pdf The dual of substitution is redecoration] (Also available as [http://www.cs.ut.ee/~varmo/papers/sfp01-book.ps.gz ps.gz].)<br />
* [http://dx.doi.org/10.1016/j.ic.2005.08.005 Recursive coalgebras from comonads]<br />
* [http://www.fing.edu.uy/~pardo/papers/njc01.ps.gz Recursion schemes from comonads]<br />
* [http://cs.ioc.ee/~tarmo/papers/essence.pdf The Essence of Dataflow Programming].<br />
<br />
=Acknowledgements=<br />
<br />
A special thanks to all of those who taught me about standard Haskell<br />
type classes and helped me develop good intuition for them,<br />
particularly Jules Bean (quicksilver), Derek Elkins (ddarius), Conal<br />
Elliott (conal), Cale Gibbard (Cale), David House, Dan Piponi<br />
(sigfpe), and Kevin Reid (kpreid).<br />
<br />
I also thank the many people who provided a mountain of helpful<br />
feedback and suggestions on a first draft of the Typeclassopedia: David Amos,<br />
Kevin Ballard, Reid Barton, Doug Beardsley, Joachim Breitner, Andrew<br />
Cave, David Christiansen, Gregory Collins, Mark Jason Dominus, Conal<br />
Elliott, Yitz Gale, George Giorgidze, Steven Grady, Travis Hartwell,<br />
Steve Hicks, Philip Hölzenspies, Edward Kmett, Eric Kow, Serge Le<br />
Huitouze, Felipe Lessa, Stefan Ljungstrand, Eric Macaulay, Rob MacAulay, Simon Meier,<br />
Eric Mertens, Tim Newsham, Russell O’Connor, Conrad Parker, Walt<br />
Rorie-Baety, Colin Ross, Tom Schrijvers, Aditya Siram, C. Smith,<br />
Martijn van Steenbergen, Joe Thornber, Jared Updike, Rob Vollmert,<br />
Andrew Wagner, Louis Wasserman, and Ashley Yakeley, as well as a few<br />
only known to me by their IRC nicks: b_jonas, maltem, tehgeekmeister,<br />
and ziman. I have undoubtedly omitted a few inadvertently, which in<br />
no way diminishes my gratitude.<br />
<br />
Finally, I would like to thank Wouter Swierstra for his fantastic work<br />
editing the Monad.Reader, and my wife Joyia for her patience during<br />
the process of writing the Typeclassopedia.<br />
<br />
=About the author=<br />
<br />
Brent Yorgey ([http://byorgey.wordpress.com/ blog], [http://www.cis.upenn.edu/~byorgey/ homepage]) is (as of November 2011) a fourth-year Ph.D. student in the [http://www.cis.upenn.edu/~plclub/ programming languages group] at the [http://www.upenn.edu University of Pennsylvania]. He enjoys teaching, creating EDSLs, playing Bach fugues, musing upon category theory, and cooking tasty lambda-treats for the denizens of #haskell.<br />
<br />
=Colophon=<br />
<br />
The Typeclassopedia was written by Brent Yorgey and initally published in March 2009. Painstakingly converted to wiki syntax by [[User:Geheimdienst]] in November 2011, after asking Brent’s permission.<br />
<br />
If something like this tex to wiki syntax conversion ever needs to be done again, here are some vim commands that helped:<br />
<br />
* <nowiki>%s/\\section{\([^}]*\)}/=\1=/gc</nowiki><br />
* <nowiki>%s/\\subsection{\([^}]*\)}/==\1==/gc</nowiki><br />
* <nowiki>%s/^ *\\item /\r* /gc</nowiki><br />
* <nowiki>%s/---/—/gc</nowiki><br />
* <nowiki>%s/\$\([^$]*\)\$/<math>\1\\ <\/math>/gc</nowiki> ''Appending “\ ” forces images to be rendered. Otherwise, Mediawiki would go back and forth between one font for short <nowiki><math></nowiki> tags, and another more Tex-like font for longer tags (containing more than a few characters)""<br />
* <nowiki>%s/|\([^|]*\)|/<code>\1<\/code>/gc</nowiki><br />
* <nowiki>%s/\\dots/.../gc</nowiki><br />
* <nowiki>%s/^\\label{.*$//gc</nowiki><br />
* <nowiki>%s/\\emph{\([^}]*\)}/''\1''/gc</nowiki><br />
* <nowiki>%s/\\term{\([^}]*\)}/''\1''/gc</nowiki><br />
<br />
The biggest issue was taking the academic-paper-style citations and turning them into hyperlinks with an appropriate title and an appropriate target. In most cases there was an obvious thing to do (e.g. online PDFs of the cited papers or Citeseer entries). Sometimes, however, it’s less clear and you might want to check the<br />
[[Media:Typeclassopedia.pdf|original Typeclassopedia PDF]]<br />
with the<br />
[http://code.haskell.org/~byorgey/TMR/Issue13/typeclassopedia.bib original bibliography file].<br />
<br />
To get all the citations into the main text, I first tried processing the source with Tex or Lyx. This didn’t work due to missing unfindable packages, syntax errors, and my general ineptitude with Tex.<br />
<br />
I then went for the next best solution, which seemed to be extracting all instances of “\cite{something}” from the source and ''in that order'' pulling the referenced entries from the .bib file. This way you can go through the source file and sorted-references file in parallel, copying over what you need, without searching back and forth in the .bib file. I used:<br />
<br />
* <nowiki>egrep -o "\cite\{[^\}]*\}" ~/typeclassopedia.lhs | cut -c 6- | tr "," "\n" | tr -d "}" > /tmp/citations</nowiki><br />
* <nowiki>for i in $(cat /tmp/citations); do grep -A99 "$i" ~/typeclassopedia.bib|egrep -B99 '^\}$' -m1 ; done > ~/typeclasso-refs-sorted</nowiki><br />
<br />
[[Category:Applicative Functor]]<br />
[[Category:Arrow]]<br />
[[Category:Functor]]<br />
[[Category:Monad]]<br />
[[Category:Standard classes]]<br />
[[Category:Standard libraries]]<br />
[[Category:Standard packages]]<br />
[[Category:Standard types]]</div>Blackouthttps://wiki.haskell.org/index.php?title=All_About_Monads&diff=54867All About Monads2012-12-07T12:51:06Z<p>Blackout: /* Example */ Changed indentation to a more dangling style to save horizontal space</p>
<hr />
<div>''All About Monads'' is a tutorial on monads and monad transformers and a walk-through of common monad instances. You can download a PDF version [http://ompldr.org/vY29zdw/All_About_Monads.pdf here] or [http://www.mediafire.com/?no3dq024hgacye4 here].<br />
<br />
Attempts are being made at porting the tutorial to this wiki; what you're seeing below is a preview of the result of that effort. If you wish to help out you should fork [https://github.com/dag/all-about-monads this GitHub repo] rather than edit this page, for now.<br />
<br />
= Introduction =<br />
<br />
== What is a monad? ==<br />
<br />
A monad is a way to structure computations in terms of values and sequences of computations using those values. Monads allow the programmer to build up computations using sequential building blocks, which can themselves be sequences of computations. The monad determines how combined computations form a new computation and frees the programmer from having to code the combination manually each time it is required.<br />
<br />
''It is useful to think of a monad as a strategy for combining computations into more complex computations.'' For example, you should be familiar with the <code>Maybe</code> type in Haskell:<br />
<br />
<haskell><br />
data Maybe a = Nothing | Just a<br />
</haskell><br />
which represents the type of computations which may fail to return a result. The <code>Maybe</code> type suggests a strategy for combining computations which return <code>Maybe</code> values: if a combined computation consists of one computation <code>B</code> that depends on the result of another computation <code>A</code>, then the combined computation should yield <code>Nothing</code> whenever either <code>A</code> or <code>B</code> yield <code>Nothing</code> and the combined computation should yield the result of <code>B</code> applied to the result of <code>A</code> when both computations succeed.<br />
<br />
Other monads exist for building computations that perform I/O, have state, may return multiple results, etc. There are as many different type of monads as there are strategies for combining computations, but there are certain monads that are especially useful and are common enough that they are part of the standard [http://www.haskell.org/onlinelibrary/ Haskell 98 libraries]. These monads are each described in [[introII.html|Part II]].<br />
<br />
== Why should I make the effort to understand monads? ==<br />
<br />
The sheer number of different [http://haskell.cs.yale.edu/bookshelf/#monads monad tutorials] on the internet is a good indication of the difficulty many people have understanding the concept. This is due to the abstract nature of monads and to the fact that they are used in several different capacities, which can confuse the picture of exactly what a monad is and what it is good for.<br />
<br />
In Haskell, monads play a central role in the I/O system. It is not essential to understand monads to do I/O in Haskell, but understanding the I/O monad will improve your code and extend your capabilities.<br />
<br />
For the programmer, monads are useful tools for structuring functional programs. They have three properties that make them especially useful:<br />
<br />
# Modularity - They allow computations to be composed from simpler computations and separate the combination strategy from the actual computations being performed.<br />
# Flexibility - They allow functional programs to be much more adaptable than equivalent programs written without monads. This is because the monad distills the computational strategy into a single place instead of requiring it be distributed throughout the entire program.<br />
# Isolation - They can be used to create imperative-style computational structures which remain safely isolated from the main body of the functional program. This is useful for incorporating side-effects (such as I/O) and state (which violates referential transparency) into a pure functional language like Haskell.<br />
<br />
Each of these features will be revisited later in the tutorial in the context of specific monads.<br />
<br />
Meet the Monads<br />
<br />
= Meet the Monads =<br />
<br />
We will use the <code>Maybe</code> type constructor throughout this chapter, so you should familiarize yourself with the definition and usage of [http://www.haskell.org/onlinelibrary/maybe.html <code>Maybe</code>] before continuing.<br />
<br />
== Type constructors ==<br />
<br />
To understand monads in Haskell, you need to be comfortable dealing with type constructors. A ''type constructor'' is a parameterized type definition used with polymorphic types. By supplying a type constructor with one or more concrete types, you can construct a new concrete type in Haskell. In the definition of <code>Maybe</code>:<br />
<br />
<haskell><br />
data Maybe a = Nothing | Just a<br />
</haskell><br />
<code>Maybe</code> is a type constructor and <code>Nothing</code> and <code>Just</code> are data constructors. You can construct a data value by applying the <code>Just</code> data constructor to a value:<br />
<br />
<haskell><br />
country = Just "China"<br />
</haskell><br />
In the same way, you can construct a type by applying the <code>Maybe</code> type constructor to a type:<br />
<br />
<haskell><br />
lookupAge :: DB -> String -> Maybe Int<br />
</haskell><br />
Polymorphic types are like containers that are capable of holding values of many different types. So <code>Maybe Int</code> can be thought of as a <code>Maybe</code> container holding an <code>Int</code> value (or <code>Nothing</code>) and <code>Maybe String</code> would be a <code>Maybe</code> container holding a <code>String</code> value (or <code>Nothing</code>). In Haskell, we can also make the type of the container polymorphic, so we could write &quot;<code>m a</code>&quot; to represent a container of some type holding a value of some type!<br />
<br />
We often use type variables with type constructors to describe abstract features of a computation. For example, the polymorphic type <code>Maybe a</code> is the type of all computations that may return a value or <code>Nothing</code>. In this way, we can talk about the properties of the container apart from any details of what the container might hold.<br />
<br />
[[Image:info.png]] If you get messages about &quot;kind errors&quot; from the compiler when working with monads, it means that you are not using the type constructors correctly. <br /><br />
<br />
<br />
== Maybe a monad ==<br />
<br />
In Haskell a monad is represented as a type constructor (call it <code>m</code>), a function that builds values of that type (<code>a -> m a</code>), and a function that combines values of that type with computations that produce values of that type to produce a new computation for values of that type (<code>m a -> (a -> m b) -> m b</code>). Note that the container is the same, but the type of the contents of the container can change. It is customary to call the monad type constructor &quot;<code>m</code>&quot; when discussing monads in general. The function that builds values of that type is traditionally called &quot;<code>return</code>&quot; and the third function is known as &quot;bind&quot; but is written &quot;<code>>>=</code>&quot;. The signatures of the functions are:<br />
<br />
<haskell><br />
-- the type of monad m<br />
data m a = ... <br />
<br />
-- return is a type constructor that creates monad instances <br />
return :: a -> m a<br />
<br />
-- bind is a function that combines a monad instance m a with a computation<br />
-- that produces another monad instance m b from a's to produce a new<br />
-- monad instance m b<br />
(>>=) :: m a -> (a -> m b) -> m b<br />
</haskell><br />
Roughly speaking, the monad type constructor defines a type of computation, the <code>return</code> function creates primitive values of that computation type and <code>>>=</code> combines computations of that type together to make more complex computations of that type. Using the container analogy, the type constructor <code>m</code> is a container that can hold different values. <code>m a</code> is a container holding a value of type <code>a</code>. The <code>return</code> function puts a value into a monad container. The <code>>>=</code> function takes the value from a monad container and passes it to a function to produce a monad container containing a new value, possibly of a different type. The <code>>>=</code> function is known as &quot;bind&quot; because it binds the value in a monad container to the first argument of a function. By adding logic to the binding function, a monad can implement a specific strategy for combining computations in the monad.<br />
<br />
This will all become clearer after the example below, but if you feel particularly confused at this point you might try looking at this [[analogy.html|physical analogy of a monad]] before continuing.<br />
<br />
== An example ==<br />
<br />
Suppose that we are writing a program to keep track of sheep cloning experiments. We would certainly want to know the genetic history of all of our sheep, so we would need <code>mother</code> and <code>father</code> functions. But since these are cloned sheep, they may not always have both a mother and a father!<br />
<br />
We would represent the possibility of not having a mother or father using the <code>Maybe</code> type constructor in our Haskell code:<br />
<br />
<haskell><br />
type Sheep = ...<br />
<br />
father :: Sheep -> Maybe Sheep<br />
father = ...<br />
<br />
mother :: Sheep -> Maybe Sheep<br />
mother = ...<br />
</haskell><br />
Then, defining functions to find grandparents is a little more complicated, because we have to handle the possibility of not having a parent:<br />
<br />
<haskell><br />
maternalGrandfather :: Sheep -> Maybe Sheep<br />
maternalGrandfather s = case (mother s) of<br />
Nothing -> Nothing<br />
Just m -> father m<br />
</haskell><br />
and so on for the other grandparent combinations.<br />
<br />
It gets even worse if we want to find great grandparents:<br />
<br />
<haskell><br />
mothersPaternalGrandfather :: Sheep -> Maybe Sheep<br />
mothersPaternalGrandfather s = case (mother s) of<br />
Nothing -> Nothing<br />
Just m -> case (father m) of<br />
Nothing -> Nothing<br />
Just gf -> father gf<br />
</haskell><br />
Aside from being ugly, unclear, and difficult to maintain, this is just too much work. It is clear that a <code>Nothing</code> value at any point in the computation will cause <code>Nothing</code> to be the final result, and it would be much nicer to implement this notion once in a single place and remove all of the explicit <code>case</code> testing scattered all over the code. This will make the code easier to write, easier to read and easier to change. So good programming style would have us create a combinator that captures the behavior we want:<br />
<br />
Code available in [[../examples/example1.hs|example1.hs]]<br />
<br />
<haskell><br />
-- comb is a combinator for sequencing operations that return Maybe<br />
comb :: Maybe a -> (a -> Maybe b) -> Maybe b<br />
comb Nothing _ = Nothing<br />
comb (Just x) f = f x<br />
<br />
-- now we can use `comb` to build complicated sequences<br />
mothersPaternalGrandfather :: Sheep -> Maybe Sheep<br />
mothersPaternalGrandfather s = (Just s) `comb` mother `comb` father `comb` father <br />
</haskell><br />
The combinator is a huge success! The code is much cleaner and easier to write, understand and modify. Notice also that the <code>comb</code> function is entirely polymorphic — it is not specialized for <code>Sheep</code> in any way. In fact, ''the combinator captures a general strategy for combining computations that may fail to return a value.'' Thus, we can apply the same combinator to other computations that may fail to return a value, such as database queries or dictionary lookups.<br />
<br />
The happy outcome is that common sense programming practice has led us to create a monad without even realizing it. The <code>Maybe</code> type constructor along with the <code>Just</code> function (acts like <code>return</code>) and our combinator (acts like <code>>>=</code>) together form a simple monad for building computations which may not return a value. All that remains to make this monad truly useful is to make it conform to the monad framework built into the Haskell language. That is the subject of the next chapter.<br />
<br />
== A list is also a monad ==<br />
<br />
We have seen that the <code>Maybe</code> type constructor is a monad for building computations which may fail to return a value. You may be surprised to know that another common Haskell type constructor, <code>[]</code> (for building lists), is also a monad. The List monad allows us to build computations which can return 0, 1, or more values.<br />
<br />
The <code>return</code> function for lists simply creates a singleton list (<code>return x = [x]</code>). The binding operation for lists creates a new list containing the results of applying the function to all of the values in the original list (<code>l >>= f = concatMap f l</code>).<br />
<br />
One use of functions which return lists is to represent ''ambiguous'' computations — that is computations which may have 0, 1, or more allowed outcomes. In a computation composed from ambigous subcomputations, the ambiguity may compound, or it may eventually resolve into a single allowed outcome or no allowed outcome at all. During this process, the set of possible computational states is represented as a list. The List monad thus embodies a strategy for performing simultaneous computations along all allowed paths of an ambiguous computation.<br />
<br />
Examples of this use of the List monad, and contrasting examples using the Maybe monad will be presented shortly. But first, we must see how useful monads are defined in Haskell.<br />
<br />
== Summary ==<br />
<br />
We have seen that a monad is a type constructor, a function called <code>return</code>, and a combinator function called <code>bind</code> or <code>>>=</code>. These three elements work together to encapsulate a strategy for combining computations to produce more complex computations.<br />
<br />
Using the <code>Maybe</code> type constructor, we saw how good programming practice led us to define a simple monad that could be used to build complex computations out of sequences of computations that could each fail to return a value. The resulting <code>Maybe</code> monad encapsulates a strategy for combining computations that may not return values. By codifying the strategy in a monad, we have achieved a degree of modularity and flexibility that is not present when the computations are combined in an ad hoc manner.<br />
<br />
We have also seen that another common Haskell type constructor, <code>[]</code>, is a monad. The List monad encapsulates a strategy for combining computations that can return 0, 1, or multiple values.<br />
<br />
Doing it with class<br />
<br />
= Doing it with class =<br />
<br />
== Haskell type classes ==<br />
<br />
The discussion in this chapter involves the Haskell type class system. If you are not familiar with type classes in Haskell, you should [http://www.haskell.org/tutorial/classes.html review them] before continuing.<br />
<br />
== The Monad class ==<br />
<br />
In Haskell, there is a standard <code>Monad</code> class that defines the names and signatures of the two monad functions <code>return</code> and <code>>>=</code>. It is not strictly necessary to make your monads instances of the <code>Monad</code> class, but it is a good idea. Haskell has special support for <code>Monad</code> instances built into the language and making your monads instances of the <code>Monad</code> class will allow you to use these features to write cleaner and more elegant code. Also, making your monads instances of the <code>Monad</code> class communicates important information to others who read the code and failing to do so can cause you to use confusing and non-standard function names. It's easy to do and it has many benefits, so just do it!<br />
<br />
The standard <code>Monad</code> class definition in Haskell looks something like this:<br />
<br />
<haskell><br />
class Monad m where<br />
(>>=) :: m a -> (a -> m b) -> m b<br />
return :: a -> m a<br />
</haskell><br />
== Example continued ==<br />
<br />
Continuing the [[meet.html#example1|previous example]], we will now see how the <code>Maybe</code> type constructor fits into the Haskell monad framework as an instance of the <code>Monad</code> class.<br />
<br />
Recall that our <code>Maybe</code> monad used the <code>Just</code> data constructor to fill the role of the monad <code>return</code> function and we built a simple combinator to fill the role of the monad <code>>>=</code> binding function. We can make its role as a monad explicit by declaring <code>Maybe</code> as an instance of the <code>Monad</code> class:<br />
<br />
<haskell><br />
instance Monad Maybe where<br />
Nothing >>= f = Nothing<br />
(Just x) >>= f = f x<br />
return = Just<br />
</haskell><br />
Once we have defined <code>Maybe</code> as an instance of the Monad class, we can use the standard monad operators to build the complex computations:<br />
<br />
<haskell><br />
-- we can use monadic operations to build complicated sequences<br />
maternalGrandfather :: Sheep -> Maybe Sheep<br />
maternalGrandfather s = (return s) >>= mother >>= father<br />
<br />
fathersMaternalGrandmother :: Sheep -> Maybe Sheep<br />
fathersMaternalGrandmother s = (return s) >>= father >>= mother >>= mother <br />
</haskell><br />
In Haskell, <code>Maybe</code> is defined as an instance of the <code>Monad</code> class in the standard prelude, so you don't need to do it yourself. The other monad we have seen so far, the list constructor, is also defined as an instance of the <code>Monad</code> class in the standard prelude.<br />
<br />
[[Image:info.png]] When writing functions that work with monads, try to make use of the <code>Monad</code> class instead of using a specific monad instance. A function of the type<br />
<br />
<haskell><br />
doSomething :: (Monad m) => a -> m b<br />
</haskell><br />
is much more flexible than one of the type<br />
<br />
<haskell><br />
doSomething :: a -> Maybe b<br />
</haskell><br />
The former function can be used with many types of monads to get different behavior depending on the strategy embodied in the monad, whereas the latter function is restricted to the strategy of the <code>Maybe</code> monad.<br />
<br />
== Do notation ==<br />
<br />
Using the standard monadic function names is good, but another advantage of membership in the <code>Monad</code> class is the Haskell support for &quot;do&quot; notation. Do notation is an expressive shorthand for building up monadic computations, similar to the way that list comprehensions are an expressive shorthand for building computations on lists. Any instance of the <code>Monad</code> class can be used in a do-block in Haskell.<br />
<br />
In short, the do notation allows you to write monadic computations using a pseudo-imperative style with named variables. The result of a monadic computation can be &quot;assigned&quot; to a variable using a left arrow <code><-</code> operator. Then using that variable in a subsequent monadic computation automatically performs the binding. The type of the expression to the right of the arrow is a monadic type <code>m a</code>. The expression to the left of the arrow is a pattern to be matched against the value ''inside'' the monad. <code>(x:xs)</code> would match against <code>Maybe [1,2,3]</code>, for example.<br />
<br />
Here is a sample of do notation using the <code>Maybe</code> monad:<br />
<br />
Code available in [[../examples/example2.hs|example2.hs]]<br />
<br />
<haskell><br />
-- we can also use do-notation to build complicated sequences<br />
mothersPaternalGrandfather :: Sheep -> Maybe Sheep<br />
mothersPaternalGrandfather s = do m <- mother s<br />
gf <- father m<br />
father gf<br />
</haskell><br />
Compare this to <code>fathersMaternalGrandmother</code> written above without using do notation.<br />
<br />
The do block shown above is written using the layout rule to define the extent of the block. Haskell also allows you to use braces and semicolons when defining a do block:<br />
<br />
<haskell><br />
mothersPaternalGrandfather s = do { m <- mother s; gf <- father m; father gf }<br />
</haskell><br />
Notice that do notation resembles an imperative programming language, in which a computation is built up from an explicit sequence of simpler computations. In this respect, monads offer the possibility to create imperative-style computations within a larger functional program. This theme will be expanded upon when we deal with side-effects and the I/O monad later.<br />
<br />
Do notation is simply syntactic sugar. There is nothing that can be done using do notation that cannot be done using only the standard monadic operators. But do notation is cleaner and more convenient in some cases, especially when the sequence of monadic computations is long. You should understand both the standard monadic binding notation and do notation and be able to apply each where they are appropriate.<br />
<br />
The actual translation from do notation to standard monadic operators is roughly that every expression matched to a pattern, <code>x <- expr1</code>, becomes<br />
<br />
<haskell><br />
expr1 >>= \x -><br />
</haskell><br />
and every expression without a variable assignment, <code>expr2</code> becomes<br />
<br />
<haskell><br />
expr2 >>= \_ -><br />
</haskell><br />
All do blocks must end with a monadic expression, and a let clause is allowed at the beginning of a do block (but let clauses in do blocks do not use the &quot;in&quot; keyword). The definition of <code>mothersPaternalGrandfather</code> above would be translated to:<br />
<br />
<haskell><br />
mothersPaternalGrandfather s = mother s >>= \m -><br />
father m >>= \gf -><br />
father gf<br />
</haskell><br />
It now becomes clear why the binding operator is so named. It is literally used to bind the value in the monad to the argument in the following lambda expression.<br />
<br />
== Summary ==<br />
<br />
Haskell provides built-in support for monads. To take advantage of Haskell's monad support, you must declare the monad type constructor to be an instance of the <code>Monad</code> class and supply definitions of the <code>return</code> and <code>>>=</code> (pronounced &quot;bind&quot;) functions for the monad.<br />
<br />
A monad that is an instance of the <code>Monad</code> class can be used with do-notation, which is syntactic sugar that provides a simple, imperative-style notation for describing computations with monads.<br />
<br />
The monad laws<br />
<br />
= The monad laws =<br />
<br />
The tutorial up to now has avoided technical discussions, but there are a few technical points that must be made concerning monads. Monadic operations must obey a set of laws, known as &quot;the monad axioms&quot;. These laws aren't enforced by the Haskell compiler, so it is up to the programmer to ensure that any <code>Monad</code> instances they declare obey the laws. Haskell's <code>Monad</code> class also includes some functions beyond the minimal complete definition that we have not seen yet. Finally, many monads obey additional laws beyond the standard monad laws, and there is an additional Haskell class to support these extended monads.<br />
<br />
== The three fundamental laws ==<br />
<br />
The concept of a monad comes from a branch of mathematics called category theory. While it is not necessary to know category theory to create and use monads, we do need to obey a small bit of mathematical formalism. To create a monad, it is not enough just to declare a Haskell instance of the <code>Monad</code> class with the correct type signatures. To be a proper monad, the <code>return</code> and <code>>>=</code> functions must work together according to three laws:<br />
<br />
# <code>(return x) >>= f == f x</code><br />
# <code>m >>= return == m</code><br />
# <code>(m >>= f) >>= g == m >>= (\x -> f x >>= g)</code><br />
<br />
The first law requires that <code>return</code> is a left-identity with respect to <code>>>=</code>. The second law requires that <code>return</code> is a right-identity with respect to <code>>>=</code>. The third law is a kind of associativity law for <code>>>=</code>. Obeying the three laws ensures that the semantics of the do-notation using the monad will be consistent.<br />
<br />
Any type constructor with return and bind operators that satisfy the three monad laws is a monad. In Haskell, the compiler does not check that the laws hold for every instance of the <code>Monad</code> class. It is up to the programmer to ensure that any <code>Monad</code> instance they create satisfies the monad laws.<br />
<br />
== Failure IS an option ==<br />
<br />
The definition of the <code>Monad</code> class given [[class.html#monad|earlier]] showed only the minimal complete definition. The full definition of the <code>Monad</code> class actually includes two additional functions: <code>fail</code> and <code>>></code>.<br />
<br />
The default implementation of the <code>fail</code> function is:<br />
<br />
<haskell><br />
fail s = error s<br />
</haskell><br />
You do not need to change this for your monad unless you want to provide different behavior for failure or to incorporate failure into the computational strategy of your monad. The <code>Maybe</code> monad, for instance, defines <code>fail</code> as:<br />
<br />
<haskell><br />
fail _ = Nothing<br />
</haskell><br />
so that <code>fail</code> returns an instance of the <code>Maybe</code> monad with meaningful behavior when it is bound with other functions in the <code>Maybe</code> monad.<br />
<br />
The <code>fail</code> function is not a required part of the mathematical definition of a monad, but it is included in the standard <code>Monad</code> class definition because of the role it plays in Haskell's do notation. The <code>fail</code> function is called whenever a pattern matching failure occurs in a do block:<br />
<br />
<haskell><br />
fn :: Int -> Maybe [Int]<br />
fn idx = do let l = [Just [1,2,3], Nothing, Just [], Just [7..20]]<br />
(x:xs) <- l!!idx -- a pattern match failure will call "fail"<br />
return xs<br />
</haskell><br />
So in the code above, <code>fn 0</code> has the value <code>Just [2,3]</code>, but <code>fn 1</code> and <code>fn 2</code> both have the value <code>Nothing</code>.<br />
<br />
The <code>>></code> function is a convenience operator that is used to bind a monadic computation that does not require input from the previous computation in the sequence. It is defined in terms of <code>>>=</code>:<br />
<br />
<haskell><br />
(>>) :: m a -> m b -> m b<br />
m >> k = m >>= (\_ -> k)<br />
</haskell><br />
== No way out ==<br />
<br />
You might have noticed that there is no way to get values out of a monad as defined in the standard <code>Monad</code> class. That is not an accident. Nothing prevents the monad author from allowing it using functions specific to the monad. For instance, values can be extracted from the <code>Maybe</code> monad by pattern matching on <code>Just x</code> or using the <code>fromJust</code> function.<br />
<br />
By not requiring such a function, the Haskell <code>Monad</code> class allows the creation of one-way monads. One-way monads allow values to enter the monad through the <code>return</code> function (and sometimes the <code>fail</code> function) and they allow computations to be performed within the monad using the bind functions <code>>>=</code> and <code>>></code>, but they do not allow values back out of the monad.<br />
<br />
The <code>IO</code> monad is a familiar example of a one-way monad in Haskell. Because you can't escape from the <code>IO</code> monad, it is impossible to write a function that does a computation in the <code>IO</code> monad but whose result type does not include the <code>IO</code> type constructor. This means that ''any'' function whose result type does not contain the <code>IO</code> type constructor is guaranteed not to use the <code>IO</code> monad. Other monads, such as <code>List</code> and <code>Maybe</code>, do allow values out of the monad. So it is possible to write functions which use these monads internally but return non-monadic values.<br />
<br />
''The wonderful feature of a one-way monad is that it can support side-effects in its monadic operations but prevent them from destroying the functional properties of the non-monadic portions of the program.''<br />
<br />
Consider the simple issue of reading a character from the user. We cannot simply have a function <code>readChar :: Char</code>, because it needs to return a different character each time it is called, depending on the input from the user. It is an essential property of Haskell as a pure functional language that all functions return the same value when called twice with the same arguments. But it ''is'' ok to have an I/O function <code>getChar :: IO Char</code> in the <code>IO</code> monad, because it can only be used in a sequence within the one-way monad. There is no way to get rid of the <code>IO</code> type constructor in the signature of any function that uses it, so the <code>IO</code> type constructor acts as a kind of tag that identifies all functions that do I/O. Furthermore, such functions are only useful within the <code>IO</code> monad. So a one-way monad effectively creates an isolated computational domain in which the rules of a pure functional language can be relaxed. Functional computations can move into the domain, but dangerous side-effects and non-referentially-transparent functions cannot escape from it.<br />
<br />
Another common pattern when defining monads is to represent monadic values as functions. Then when the value of a monadic computation is required, the resulting monad is &quot;run&quot; to provide the answer.<br />
<br />
== Zero and Plus ==<br />
<br />
Beyond the three monad laws stated above, some monads obey additional laws. The monads have a special value <code>mzero</code> and an operator <code>mplus</code> that obey four additional laws:<br />
<br />
# <code>mzero >>= f == mzero</code><br />
# <code>m >>= (\x -> mzero) == mzero</code><br />
# <code>mzero `mplus` m == m</code><br />
# <code>m `mplus` mzero == m</code><br />
<br />
It is easy to remember the laws for <code>mzero</code> and <code>mplus</code> if you associate <code>mzero</code> with 0, <code>mplus</code> with +, and <code>>>=</code> with × in ordinary arithmetic.<br />
<br />
Monads which have a zero and a plus can be declared as instances of the <code>MonadPlus</code> class in Haskell:<br />
<br />
<haskell><br />
class (Monad m) => MonadPlus m where<br />
mzero :: m a<br />
mplus :: m a -> m a -> m a<br />
</haskell><br />
Continuing to use the <code>Maybe</code> monad as an example, we see that the <code>Maybe</code> monad is an instance of <code>MonadPlus</code>:<br />
<br />
<haskell><br />
instance MonadPlus Maybe where<br />
mzero = Nothing<br />
Nothing `mplus` x = x<br />
x `mplus` _ = x<br />
</haskell><br />
This identifies <code>Nothing</code> as the zero value and says that adding two <code>Maybe</code> values together gives the first value that is not <code>Nothing</code>. If both input values are <code>Nothing</code>, then the result of <code>mplus</code> is also <code>Nothing</code>.<br />
<br />
The List monad also has a zero and a plus. <code>mzero</code> is the empty list and <code>mplus</code> is the <code>++</code> operator.<br />
<br />
The <code>mplus</code> operator is used to combine monadic values from separate computations into a single monadic value. Within the context of our sheep-cloning example, we could use <code>Maybe</code>'s <code>mplus</code> to define a function, <code>parent s = (mother s) `mplus` (father s)</code>, which would return a parent if there is one, and <code>Nothing</code> is the sheep has no parents at all. For a sheep with both parents, the function would return one or the other, depending on the exact definition of <code>mplus</code> in the <code>Maybe</code> monad.<br />
<br />
== Summary ==<br />
<br />
Instances of the <code>Monad</code> class should conform to the so-called monad laws, which describe algabraic properties of monads. There are three of these laws which state that the <code>return</code> function is both a left and a right identity and that the binding operator is associative. Failure to satisfy these laws will result in monads that do not behave properly and may cause subtle problems when using do-notation.<br />
<br />
In addition to the <code>return</code> and <code>>>=</code> functions, the <code>Monad</code> class defines another function, <code>fail</code>. The <code>fail</code> function is not a technical requirement for inclusion as a monad, but it is often useful in practice and it is included in the <code>Monad</code> class because it is used in Haskell's do-notation.<br />
<br />
Some monads obey laws beyond the three basic monad laws. An important class of such monads are ones which have a notion of a zero element and a plus operator. Haskell provides a <code>MonadPlus</code> class for such monads which define the <code>mzero</code> value and the <code>mplus</code> operator.<br />
<br />
Exercises<br />
<br />
= Exercises =<br />
<br />
# [[#exercise1|Do notation]]<br />
# [[#exercise2|Combining monadic values]]<br />
# [[#exercise3|Using the List monad]]<br />
# [[#exercise4|Using the Monad class constraint]]<br />
<br />
This section contains a few simple exercises to hone the reader's monadic reasoning skills and to provide a solid comprehension of the function and use of the Maybe and List monads before looking at monadic programming in more depth. The exercises will build on the previous sheep-cloning [[../examples/example2.hs|example]], with which the reader should already be familiar.<br />
<br />
== Exercise 1: Do notation ==<br />
<br />
Rewrite the <code>maternalGrandfather</code>, <code>fathersMaternalGrandmother</code>, and <code>mothersPaternalGrandfather</code> functions in [[../examples/example2.hs|Example 2]] using the monadic operators <code>return</code> and <code>>>=</code>, without using any do-notation syntactic sugar.<br />
<br />
<br />
<br />
Click [[solution1.html|here]] to see the solution.<br />
<br />
== Exercise 2: Combining monadic values ==<br />
<br />
Write functions <code>parent</code> and <code>grandparent</code> with signature <code>Sheep -> Maybe Sheep</code>. They should return one sheep selected from all sheep matching the description, or <code>Nothing</code> if there is no such sheep. Hint: the <code>mplus</code> operator is useful here.<br />
<br />
Click [[solution2.html|here]] to see the solution.<br />
<br />
== Exercise 3: Using the List monad ==<br />
<br />
Write functions <code>parent</code> and <code>grandparent</code> with signature <code>Sheep -> [Sheep]</code>. They should return all sheep matching the description, or the empty list if there is no such sheep. Hint: the <code>mplus</code> operator in the List monad is useful here. Also the <code>maybeToList</code> function in the <code>Maybe</code> module can be used to convert a value from the Maybe monad to the List monad.<br />
<br />
Click [[solution3.html|here]] to see the solution.<br />
<br />
== Exercise 4: Using the Monad class constraint ==<br />
<br />
Monads promote modularity and code reuse by encapsulating often-used computational strategies into single blocks of code that can be used to construct many different computations. Less obviously, monads also promote modularity by allowing you to vary the monad in which a computation is done to achieve different variations of the computation. This is achieved by writing functions which are polymorphic in the monad type constructor, using the <code>(Monad m) =></code>, <code>(MonadPlus m) =></code>, etc. class constraints.<br />
<br />
Write functions <code>parent</code> and <code>grandparent</code> with signature <code>(MonadPlus m) => Sheep -> m Sheep</code>. They should be useful in both the Maybe and List monads. How does the functions' behavior differ when used with the List monad versus the Maybe monad? If you need to review the use of type classes and class constraints in Haskell, look [http://www.haskell.org/tutorial/classes.html here].<br />
<br />
Click [[solution4.html|here]] to see the solution.<br />
<br />
Monad support in Haskell<br />
<br />
= Monad support in Haskell =<br />
<br />
Haskell's built in support for monads is split among the standard prelude, which exports the most common monad functions, and the Monad module, which contains less-commonly used monad functions. The individual monad types are each in their own libraries and are the subject of [[introII.html|Part II]] of this tutorial.<br />
<br />
== In the standard prelude ==<br />
<br />
The Haskell 98 [http://www.haskell.org/onlinelibrary/standard-prelude.html standard prelude] includes the definition of the <code>Monad</code> class as well as a few auxilliary functions for working with monadic data types.<br />
<br />
=== The <code>Monad</code> class ===<br />
<br />
We have seen the <code>Monad</code> class before:<br />
<br />
<haskell><br />
class Monad m where<br />
(>>=) :: m a -> (a -> m b) -> m b<br />
(>>) :: m a -> m b -> m b<br />
return :: a -> m a<br />
fail :: String -> m a<br />
<br />
-- Minimal complete definition:<br />
-- (>>=), return<br />
m >> k = m >>= \_ -> k<br />
fail s = error s<br />
</haskell><br />
=== The sequencing functions ===<br />
<br />
The <code>sequence</code> function takes a list of monadic computations, executes each one in turn and returns a list of the results. If any of the computations fail, then the whole function fails:<br />
<br />
<haskell><br />
sequence :: Monad m => [m a] -> m [a] <br />
sequence = foldr mcons (return [])<br />
where mcons p q = p >>= \x -> q >>= \y -> return (x:y)<br />
</haskell><br />
The <code>sequence_</code> function (notice the underscore) has the same behavior as <code>sequence</code> but does not return a list of results. It is useful when only the side-effects of the monadic computations are important.<br />
<br />
<haskell><br />
sequence_ :: Monad m => [m a] -> m () <br />
sequence_ = foldr (>>) (return ())<br />
</haskell><br />
=== The mapping functions ===<br />
<br />
The <code>mapM</code> function maps a monadic computation over a list of values and returns a list of the results. It is defined in terms of the list <code>map</code> function and the <code>sequence</code> function above:<br />
<br />
<haskell><br />
mapM :: Monad m => (a -> m b) -> [a] -> m [b] <br />
mapM f as = sequence (map f as)<br />
</haskell><br />
There is also a version with an underscore, <code>mapM_</code> which is defined using sequence_. <code>mapM_</code> operates the same as <code>mapM</code>, but it doesn't return the list of values. It is useful when only the side-effects of the monadic computation are important.<br />
<br />
<haskell><br />
mapM_ :: Monad m => (a -> m b) -> [a] -> m () <br />
mapM_ f as = sequence_ (map f as)<br />
</haskell><br />
As a simple example of the use the mapping functions, a <code>putString</code> function for the <code>IO</code> monad could be defined as:<br />
<br />
<haskell><br />
putString :: [Char] -> IO ()<br />
putString s = mapM_ putChar s<br />
</haskell><br />
<code>mapM</code> can be used within a do block in a manner similar to the way the <code>map</code> function is normally used on lists. This is a common pattern with monads — a version of a function for use within a monad (i.e., intended for binding) will have a signature similar to the non-monadic version but the function outputs will be within the monad:<br />
<br />
<haskell><br />
-- compare the non-monadic and monadic signatures<br />
map :: (a -> b) -> [a] -> [b]<br />
mapM :: Monad m => (a -> m b) -> [a] -> m [b] <br />
</haskell><br />
=== The reverse binder function (<code>=<<</code>) ===<br />
<br />
The prelude also defines a binding function that takes it arguments in the opposite order to the standard binding function. Since the standard binding function is called &quot;<code>>>=</code>&quot;, the reverse binding function is called &quot;<code>=<<</code>&quot;. It is useful in circumstances where the binding operator is used as a higher-order term and it is more convenient to have the arguments in the reversed order. Its definition is simply:<br />
<br />
<haskell><br />
(=<<) :: Monad m => (a -> m b) -> m a -> m b<br />
f =<< x = x >>= f<br />
</haskell><br />
== In the Monad module ==<br />
<br />
The <code>Monad</code> module in the standard Haskell 98 libraries exports a number of facilities for more advanced monadic operations. To access these facilities, simply <code>import Monad</code> in your Haskell program.<br />
<br />
Not all of the function in the <code>Monad</code> module are discussed here, but you are encouraged to [http://www.haskell.org/onlinelibrary/monad.html explore the module for yourself] when you feel you are ready to see some of the more esoteric monad functions.<br />
<br />
=== The <code>MonadPlus</code> class ===<br />
<br />
The <code>Monad</code> module defines the <code>MonadPlus</code> class for monads with a zero element and a plus operator:<br />
<br />
<haskell><br />
class Monad m => MonadPlus m where<br />
mzero :: m a<br />
mplus :: m a -> m a -> m a<br />
</haskell><br />
=== Monadic versions of list functions ===<br />
<br />
Several functions are provided which generalize standard list-processing functions to monads. The <code>mapM</code> functions are exported in the standard prelude and were described above.<br />
<br />
<code>foldM</code> is a monadic version of <code>foldl</code> in which monadic computations built from a list are bound left-to-right. The definition is:<br />
<br />
<haskell><br />
foldM :: (Monad m) => (a -> b -> m a) -> a -> [b] -> m a<br />
foldM f a [] = return a<br />
foldM f a (x:xs) = f a x >>= \y -> foldM f y xs<br />
</haskell><br />
but it is easier to understand the operation of <code>foldM</code> if you consider its effect in terms of a do block:<br />
<br />
<haskell><br />
-- this is not valid Haskell code, it is just for illustration<br />
foldM f a1 [x1,x2,...,xn] = do a2 <- f a1 x1<br />
a3 <- f a2 x2<br />
...<br />
f an xn<br />
</haskell><br />
Right-to-left binding is achieved by reversing the input list before calling <code>foldM</code>.<br />
<br />
We can use <code>foldM</code> to create a more poweful query function in our sheep cloning example:<br />
<br />
Code available in [[../examples/example3.hs|example3.hs]]<br />
<br />
<haskell><br />
-- traceFamily is a generic function to find an ancestor<br />
traceFamily :: Sheep -> [ (Sheep -> Maybe Sheep) ] -> Maybe Sheep<br />
traceFamily s l = foldM getParent s l<br />
where getParent s f = f s<br />
<br />
-- we can define complex queries using traceFamily in an easy, clear way<br />
mothersPaternalGrandfather s = traceFamily s [mother, father, father]<br />
paternalGrandmother s = traceFamily s [father, mother]<br />
</haskell><br />
The <code>traceFamily</code> function uses <code>foldM</code> to create a simple way to trace back in the family tree to any depth and in any pattern. In fact, it is probably clearer to write &quot;<code>traceFamily s [father, mother]</code>&quot; than it is to use the <code>paternalGrandmother</code> function!<br />
<br />
A more typical use of <code>foldM</code> is within a do block:<br />
<br />
Code available in [[../examples/example4.hs|example4.hs]]<br />
<br />
<haskell><br />
-- a Dict is just a finite map from strings to strings<br />
type Dict = FiniteMap String String<br />
<br />
-- this an auxilliary function used with foldl<br />
addEntry :: Dict -> Entry -> Dict<br />
addEntry d e = addToFM d (key e) (value e)<br />
<br />
-- this is an auxiliiary function used with foldM inside the IO monad<br />
addDataFromFile :: Dict -> Handle -> IO Dict<br />
addDataFromFile dict hdl = do contents <- hGetContents hdl<br />
entries <- return (map read (lines contents))<br />
return (foldl (addEntry) dict entries)<br />
<br />
-- this program builds a dictionary from the entries in all files named on the<br />
-- command line and then prints it out as an association list<br />
main :: IO ()<br />
main = do files <- getArgs<br />
handles <- mapM openForReading files<br />
dict <- foldM addDataFromFile emptyFM handles<br />
print (fmToList dict)<br />
</haskell><br />
The <code>filterM</code> function works like the list <code>filter</code> function inside of a monad. It takes a predicate function which returns a Boolean value in the monad and a list of values. It returns, inside the monad, a list of those values for which the predicate was True.<br />
<br />
<haskell><br />
filterM :: Monad m => (a -> m Bool) -> [a] -> m [a]<br />
filterM p [] = return []<br />
filterM p (x:xs) = do b <- p x<br />
ys <- filterM p xs<br />
return (if b then (x:ys) else ys)<br />
</haskell><br />
Here is an example showing how <code>filterM</code> can be used within the <code>IO</code> monad to select only the directories from a list:<br />
<br />
Code available in [[../examples/example5.hs|example5.hs]]<br />
<br />
<haskell><br />
import Monad<br />
import Directory<br />
import System<br />
<br />
-- NOTE: doesDirectoryExist has type FilePath -> IO Bool<br />
<br />
-- this program prints only the directories named on the command line<br />
main :: IO ()<br />
main = do names <- getArgs<br />
dirs <- filterM doesDirectoryExist names<br />
mapM_ putStrLn dirs<br />
</haskell><br />
<code>zipWithM</code> is a monadic version of the <code>zipWith</code> function on lists. <code>zipWithM_</code> behaves the same but discards the output of the function. It is useful when only the side-effects of the monadic computation matter.<br />
<br />
<haskell><br />
zipWithM ::(Monad m) => (a -> b -> m c) -> [a] -> [b] -> m [c]<br />
zipWithM f xs ys = sequence (zipWith f xs ys)<br />
<br />
zipWithM_ ::(Monad m) => (a -> b -> m c) -> [a] -> [b] -> m ()<br />
zipWithM_ f xs ys = sequence_ (zipWith f xs ys)<br />
</haskell><br />
=== Conditional monadic computations ===<br />
<br />
There are two functions provided for conditionally executing monadic computations. The <code>when</code> function takes a boolean argument and a monadic computation with unit &quot;()&quot; type and performs the computation only when the boolean argument is <code>True</code>. The <code>unless</code> function does the same, except that it performs the computation ''unless'' the boolean argument is <code>True</code>.<br />
<br />
<haskell><br />
when :: (Monad m) => Bool -> m () -> m ()<br />
when p s = if p then s else return ()<br />
<br />
unless :: (Monad m) => Bool -> m () -> m ()<br />
unless p s = when (not p) s<br />
</haskell><br />
=== <code>ap</code> and the lifting functions ===<br />
<br />
''Lifting'' is a monadic operation that converts a non-monadic function into an equivalent function that operates on monadic values. We say that a function is &quot;lifted into the monad&quot; by the lifting operators. A lifted function is useful for operating on monad values outside of a do block and can also allow for cleaner code within a do block.<br />
<br />
The simplest lifting operator is <code>liftM</code>, which lifts a function of a single argument into a monad.<br />
<br />
<haskell><br />
liftM :: (Monad m) => (a -> b) -> (m a -> m b)<br />
liftM f = \a -> do { a' <- a; return (f a') }<br />
</haskell><br />
Lifting operators are also provided for functions with more arguments. <code>liftM2</code> lifts functions of two arguments:<br />
<br />
<haskell><br />
liftM2 :: (Monad m) => (a -> b -> c) -> (m a -> m b -> m c)<br />
liftM2 f = \a b -> do { a' <- a; b' <- b; return (f a' b') }<br />
</haskell><br />
The same pattern is applied to give the definitions to lift functions of more arguments. Functions up to <code>liftM5</code> are defined in the <code>Monad</code> module.<br />
<br />
To see how the lifting operators allow more concise code, consider a computation in the <code>Maybe</code> monad in which you want to use a function <code>swapNames::String -> String</code>. You could do:<br />
<br />
<haskell><br />
getName :: String -> Maybe String<br />
getName name = do let db = [("John", "Smith, John"), ("Mike", "Caine, Michael")]<br />
tempName <- lookup name db<br />
return (swapNames tempName)<br />
</haskell><br />
But making use of the <code>liftM</code> function, we can use <code>liftM swapNames</code> as a function of type <code>Maybe String -> Maybe String</code>:<br />
<br />
Code available in [[../examples/example6.hs|example6.hs]]<br />
<br />
<haskell><br />
getName :: String -> Maybe String<br />
getName name = do let db = [("John", "Smith, John"), ("Mike", "Caine, Michael")]<br />
liftM swapNames (lookup name db)<br />
</haskell><br />
The difference is even greater when lifting functions with more arguments.<br />
<br />
The lifting functions also enable very concise constructions using higher-order functions. To understand this example code, you might need to review the definition of the monad functions for the [[listmonad.html#definition|List monad]] (particularly <code>>>=</code>). Imagine how you might implement this function without lifting the operator:<br />
<br />
Code available in [[../examples/example7.hs|example7.hs]]<br />
<br />
<haskell><br />
-- allCombinations returns a list containing the result of<br />
-- folding the binary operator through all combinations<br />
-- of elements of the given lists<br />
-- For example, allCombinations (+) [[0,1],[1,2,3]] would be<br />
-- [0+1,0+2,0+3,1+1,1+2,1+3], or [1,2,3,2,3,4]<br />
-- and allCombinations (*) [[0,1],[1,2],[3,5]] would be<br />
-- [0*1*3,0*1*5,0*2*3,0*2*5,1*1*3,1*1*5,1*2*3,1*2*5], or [0,0,0,0,3,5,6,10]<br />
allCombinations :: (a -> a -> a) -> [[a]] -> [a]<br />
allCombinations fn [] = []<br />
allCombinations fn (l:ls) = foldl (liftM2 fn) l ls<br />
</haskell><br />
There is a related function called <code>ap</code> that is sometimes more convenient to use than the lifting functions. <code>ap</code> is simply the function application operator (<code>$</code>) lifted into the monad:<br />
<br />
<haskell><br />
ap :: (Monad m) => m (a -> b) -> m a -> m b<br />
ap = liftM2 ($)<br />
</haskell><br />
Note that <code>liftM2 f x y</code> is equivalent to <code>return f `ap` x `ap` y</code>, and so on for functions of more arguments. <code>ap</code> is useful when working with higher-order functions and monads.<br />
<br />
The effect of <code>ap</code> depends on the strategy of the monad in which it is used. So for example <code>[(*2),(+3)] `ap` [0,1,2]</code> is equal to <code>[0,2,4,3,4,5]</code> and <code>(Just (*2)) `ap` (Just 3)</code> is <code>Just 6</code>. Here is a simple example that shows how <code>ap</code> can be useful when doing higher-order computations:<br />
<br />
Code available in [[../examples/example8.hs|example8.hs]]<br />
<br />
<haskell><br />
-- lookup the commands and fold ap into the command list to<br />
-- compute a result.<br />
main :: IO ()<br />
main = do let fns = [("double",(2*)), ("halve",(`div`2)),<br />
("square",(\x->x*x)), ("negate", negate),<br />
("incr",(+1)), ("decr",(+(-1)))<br />
]<br />
args <- getArgs<br />
let val = read (args!!0)<br />
cmds = map ((flip lookup) fns) (words (args!!1))<br />
print $ foldl (flip ap) (Just val) cmds<br />
</haskell><br />
=== Functions for use with <code>MonadPlus</code> ===<br />
<br />
There are two functions in the <code>Monad</code> module that are used with monads that have a zero and a plus. The first function is <code>msum</code>, which is analogous to the <code>sum</code> function on lists of integers. <code>msum</code> operates on lists of monadic values and folds the <code>mplus</code> operator into the list using the <code>mzero</code> element as the initial value:<br />
<br />
<haskell><br />
msum :: MonadPlus m => [m a] -> m a<br />
msum xs = foldr mplus mzero xs<br />
</haskell><br />
In the List monad, <code>msum</code> is equivalent to <code>concat</code>. In the <code>Maybe</code> monad, <code>msum</code> returns the first non-<code>Nothing</code> value from a list. Likewise, the behavior in other monads will depend on the exact nature of their <code>mzero</code> and <code>mplus</code> definitions.<br />
<br />
<code>msum</code> allows many recursive functions and folds to be expressed more concisely. In the <code>Maybe</code> monad, for example, we can write:<br />
<br />
Code available in [[../examples/example9.hs|example9.hs]]<br />
<br />
<haskell><br />
type Variable = String<br />
type Value = String<br />
type EnvironmentStack = [[(Variable,Value)]]<br />
<br />
-- lookupVar retrieves a variable's value from the environment stack<br />
-- It uses msum in the Maybe monad to return the first non-Nothing value.<br />
lookupVar :: Variable -> EnvironmentStack -> Maybe Value<br />
lookupVar var stack = msum $ map (lookup var) stack<br />
</haskell><br />
instead of:<br />
<br />
<haskell><br />
lookupVar :: Variable -> EnvironmentStack -> Maybe Value<br />
lookupVar var [] = Nothing<br />
lookupVar var (e:es) = let val = lookup var e<br />
in maybe (lookupVar var es) Just val<br />
</haskell><br />
The second function for use with monads with a zero and a plus is the <code>guard</code> function:<br />
<br />
<haskell><br />
guard :: MonadPlus m => Bool -> m ()<br />
guard p = if p then return () else mzero<br />
</haskell><br />
The trick to understanding this function is to recall the law for monads with zero and plus that states <code>mzero >>= f == mzero</code>. So, placing a <code>guard</code> function in a sequence of monadic operations will force any execution in which the guard is <code>False</code> to be <code>mzero</code>. This is similar to the way that guard predicates in a list comprehension cause values that fail the predicate to become <code>[]</code>.<br />
<br />
Here is an example demonstrating the use of the <code>guard</code> function in the <code>Maybe</code> monad.<br />
<br />
Code available in [[../examples/example10.hs|example10.hs]]<br />
<br />
<haskell><br />
data Record = Rec {name::String, age::Int} deriving Show<br />
type DB = [Record]<br />
<br />
-- getYoungerThan returns all records for people younger than a specified age.<br />
-- It uses the guard function to eliminate records for ages at or over the limit.<br />
-- This is just for demonstration purposes. In real life, it would be<br />
-- clearer to simply use filter. When the filter criteria are more complex,<br />
-- guard becomes more useful.<br />
getYoungerThan :: Int -> DB -> [Record]<br />
getYoungerThan limit db = mapMaybe (\r -> do { guard (age r < limit); return r }) db<br />
</haskell><br />
== Summary ==<br />
<br />
Haskell provides a number of functions which are useful for working with monads in the standard libraries. The <code>Monad</code> class and most common monad functions are in the standard prelude. The <code>MonadPlus</code> class and less commonly-used (but still very useful!) functions are defined in the <code>Monad</code> module. Many other types in the Haskell libraries are declared as instances of <code>Monad</code> and <code>MonadPlus</code> in their respective modules.<br />
<br />
Part II - Introduction<br />
<br />
= Introduction =<br />
<br />
The monads covered in Part II include a mixture of standard Haskell types that are monads as well as monad classes from Andy Gill's Monad Template Library. The Monad Template Library is included in the Glasgow Haskell Compiler's hierarchical libraries under [http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.html Control.Monad]<br />
<br />
Some of the documentation for these monads comes from the excellent [http://www.haskell.org/hawiki Haskell Wiki]. In addition to the monads covered here, monads appear many other places in Haskell, such as the [http://www.cs.uu.nl/people/daan/papers/parsec.html Parsec] monadic combinator parsing library. These monads are beyond the scope of this reference, but they are thoroughly documented on their own. You can get a taste of the Parsec library by looking in the [[../examples/example16.hs|source code]] for [[readermonad.html#example|example 16]].<br />
<br />
<table><br />
<thead><br />
<tr class="header"><br />
<th align="left">Monad</th><br />
<th align="left">Type of computation</th><br />
<th align="left">Combination strategy for <code>>>=</code></th><br />
</tr><br />
</thead><br />
<tbody><br />
<tr class="odd"><br />
<td align="left">[[identitymonad.html|Identity]]</td><br />
<td align="left">''N/A — Used with monad transformers''</td><br />
<td align="left">The bound function is applied to the input value.</td><br />
</tr><br />
<tr class="even"><br />
<td align="left">[[maybemonad.html|Maybe]]</td><br />
<td align="left">Computations which may not return a result</td><br />
<td align="left"><code>Nothing</code> input gives <code>Nothing</code> output<br /><br />
<code>Just x</code> input uses <code>x</code> as input to the bound function.</td><br />
</tr><br />
<tr class="odd"><br />
<td align="left">[[errormonad.html|Error]]</td><br />
<td align="left">Computations which can fail or throw exceptions</td><br />
<td align="left">Failure records information describing the failure. Binding passes failure information on without executing the bound function, or uses successful values as input to the bound function.</td><br />
</tr><br />
<tr class="even"><br />
<td align="left">[[listmonad.html|[] (List)]]</td><br />
<td align="left">Non-deterministic computations which can return multiple possible results</td><br />
<td align="left">Maps the bound function onto the input list and concatenates the resulting lists to get a list of all possible results from all possible inputs.</td><br />
</tr><br />
<tr class="odd"><br />
<td align="left">[[iomonad.html|IO]]</td><br />
<td align="left">Computations which perform I/O</td><br />
<td align="left">Sequential execution of I/O actions in the order of binding.</td><br />
</tr><br />
<tr class="even"><br />
<td align="left">[[statemonad.html|State]]</td><br />
<td align="left">Computations which maintain state</td><br />
<td align="left">The bound function is applied to the input value to produce a state transition function which is applied to the input state.</td><br />
</tr><br />
<tr class="odd"><br />
<td align="left">[[readermonad.html|Reader]]</td><br />
<td align="left">Computations which read from a shared environment</td><br />
<td align="left">The bound function is applied to the value of the input using the same environment.</td><br />
</tr><br />
<tr class="even"><br />
<td align="left">[[writermonad.html|Writer]]</td><br />
<td align="left">Computations which write data in addition to computing values</td><br />
<td align="left">Written data is maintained separately from values. The bound function is applied to the input value and anything it writes is appended to the write data stream.</td><br />
</tr><br />
<tr class="odd"><br />
<td align="left">[[contmonad.html|Cont]]</td><br />
<td align="left">Computations which can be interrupted and restarted</td><br />
<td align="left">The bound function is inserted into the continuation chain.</td><br />
</tr><br />
</tbody><br />
</table><br />
<br />
The Identity monad<br />
<br />
= The Identity monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Simple function application<br />
<br />
Binding strategy:<br />
<br />
The bound function is applied to the input value. <code>Identity x >>= f == Identity (f x)</code><br />
<br />
Useful for:<br />
<br />
Monads can be derived from monad transformers applied to the Identity monad.<br />
<br />
Zero and plus:<br />
<br />
None.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.Identity.html Identity a]<br />
<br />
== Motivation ==<br />
<br />
The Identity monad is a monad that does not embody any computational strategy. It simply applies the bound function to its input without any modification. Computationally, there is no reason to use the Identity monad instead of the much simpler act of simply applying functions to their arguments. The purpose of the Identity monad is its fundamental role in the theory of monad transformers (covered in Part III). Any monad transformer applied to the Identity monad yields a non-transformer version of that monad.<br />
<br />
== Definition ==<br />
<br />
<haskell><br />
newtype Identity a = Identity { runIdentity :: a } <br />
<br />
instance Monad Identity where<br />
return a = Identity a -- i.e. return = id <br />
(Identity x) >>= f = f x -- i.e. x >>= f = f x <br />
</haskell><br />
The <code>runIdentity</code> label is used in the type definition because it follows a style of monad definition that explicitly represents monad values as computations. In this style, a monadic computation is built up using the monadic operators and then the value of the computation is extracted using the <code>run******</code> function. Because the Identity monad does not do any computation, its definition is trivial. For a better example of this style of monad, see the [[statemonad.html|State]] monad.<br />
<br />
== Example ==<br />
<br />
A typical use of the Identity monad is to derive a monad from a monad transformer.<br />
<br />
<haskell><br />
-- derive the State monad using the StateT monad transformer<br />
type State s a = StateT s Identity a<br />
</haskell><br />
The Maybe monad<br />
<br />
= The Maybe monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which may return <code>Nothing</code><br />
<br />
Binding strategy:<br />
<br />
<code>Nothing</code> values bypass the bound function, other values are used as inputs to the bound function.<br />
<br />
Useful for:<br />
<br />
Building computations from sequences of functions that may return <code>Nothing</code>. Complex database queries or dictionary lookups are good examples.<br />
<br />
Zero and plus:<br />
<br />
<code>Nothing</code> is the zero. The plus operation returns the first non-<code>Nothing</code> value or <code>Nothing</code> is both inputs are <code>Nothing</code>.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/onlinelibrary/maybe.html Maybe a]<br />
<br />
== Motivation ==<br />
<br />
The Maybe monad embodies the strategy of combining a chain of computations that may each return <code>Nothing</code> by ending the chain early if any step produces <code>Nothing</code> as output. It is useful when a computation entails a sequence of steps that depend on one another, and in which some steps may fail to return a value.<br />
<br />
[[Image:info.png]] If you ever find yourself writing code like this:<br /><br />
<br />
<br />
<haskell><br />
case ... of<br />
Nothing -> Nothing<br />
Just x -> case ... of<br />
Nothing -> Nothing<br />
Just y -> ...<br />
</haskell><br />
you should consider using the monadic properties of <code>Maybe</code> to improve the code.<br />
<br />
== Definition ==<br />
<br />
<haskell><br />
data Maybe a = Nothing | Just a<br />
<br />
instance Monad Maybe where<br />
return = Just<br />
fail = Nothing<br />
Nothing >>= f = Nothing<br />
(Just x) >>= f = f x<br />
<br />
instance MonadPlus Maybe where<br />
mzero = Nothing<br />
Nothing `mplus` x = x<br />
x `mplus` _ = x<br />
</haskell><br />
== Example ==<br />
<br />
A common example is in combining dictionary lookups. Given a dictionary that maps full names to email addresses, another that maps nicknames to email addresses, and a third that maps email addresses to email preferences, you could create a function that finds a person's email preferences based on either a full name or a nickname.<br />
<br />
Code available in [[../examples/example11.hs|example11.hs]]<br />
<br />
<haskell><br />
data MailPref = HTML | Plain<br />
data MailSystem = ...<br />
<br />
getMailPrefs :: MailSystem -> String -> Maybe MailPref<br />
getMailPrefs sys name =<br />
do let nameDB = fullNameDB sys<br />
nickDB = nickNameDB sys<br />
prefDB = prefsDB sys<br />
addr <- (lookup name nameDB) `mplus` (lookup name nickDB)<br />
lookup addr prefDB<br />
</haskell><br />
The Error monad<br />
<br />
= The Error monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which may fail or throw exceptions<br />
<br />
Binding strategy:<br />
<br />
Failure records information about the cause/location of the failure. Failure values bypass the bound function, other values are used as inputs to the bound function.<br />
<br />
Useful for:<br />
<br />
Building computations from sequences of functions that may fail or using exception handling to structure error handling.<br />
<br />
Zero and plus:<br />
<br />
Zero is represented by an empty error and the plus operation executes its second argument if the first fails.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/onlinelibrary/standard-prelude.html#$tEither Either String a]<br />
<br />
== Motivation ==<br />
<br />
The Error monad (also called the Exception monad) embodies the strategy of combining computations that can throw exceptions by bypassing bound functions from the point an exception is thrown to the point that it is handled.<br />
<br />
The [http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.Error.html <code>MonadError</code>] class is parameterized over the type of error information and the monad type constructor. It is common to use <code>Either String</code> as the monad type constructor for an error monad in which error descriptions take the form of strings. In that case and many other common cases the resulting monad is already defined as an instance of the <code>MonadError</code> class. You can also define your own error type and/or use a monad type constructor other than <code>Either String</code> or <code>Either IOError</code>. In these cases you will have to explicitly define instances of the <code>Error</code> and/or <code>MonadError</code> classes.<br />
<br />
== Definition ==<br />
<br />
The definition of the <code>MonadError</code> class below uses multi-parameter type classes and funDeps, which are language extensions not found in standard Haskell 98. You don't need to understand them to take advantage of the <code>MonadError</code> class.<br />
<br />
<haskell><br />
class Error a where<br />
noMsg :: a<br />
strMsg :: String -> a<br />
<br />
class (Monad m) => MonadError e m | m -> e where<br />
throwError :: e -> m a<br />
catchError :: m a -> (e -> m a) -> m a<br />
</haskell><br />
<code>throwError</code> is used within a monadic computation to begin exception processing. <code>catchError</code> provides a handler function to handle previous errors and return to normal execution. A common idiom is:<br />
<br />
<haskell><br />
do { action1; action2; action3 } `catchError` handler <br />
</haskell><br />
where the <code>action</code> functions can call <code>throwError</code>. Note that <code>handler</code> and the do-block must have the same return type.<br />
<br />
The definition of the <code>Either e</code> type constructor as an instance of the <code>MonadError</code> class is straightforward. Following convention, <code>Left</code> is used for error values and <code>Right</code> is used for non-error (right) values.<br />
<br />
<br />
<br />
<haskell><br />
instance MonadError (Either e) where <br />
throwError = Left <br />
(Left e) `catchError` handler = handler e <br />
a `catchError` _ = a <br />
</haskell><br />
== Example ==<br />
<br />
Here is an example that demonstrates the use of a custom <code>Error</code> data type with the <code>ErrorMonad</code>'s <code>throwError</code> and <code>catchError</code> exception mechanism. The example attempts to parse hexadecimal numbers and throws an exception if an invalid character is encountered. We use a custom <code>Error</code> data type to record the location of the parse error. The exception is caught by a calling function and handled by printing an informative error message.<br />
<br />
Code available in [[../examples/example12.hs|example12.hs]]<br />
<br />
<haskell><br />
-- This is the type of our parse error representation.<br />
data ParseError = Err {location::Int, reason::String}<br />
<br />
-- We make it an instance of the Error class<br />
instance Error ParseError where<br />
noMsg = Err 0 "Parse Error"<br />
strMsg s = Err 0 s<br />
<br />
-- For our monad type constructor, we use Either ParseError<br />
-- which represents failure using Left ParseError or a<br />
-- successful result of type a using Right a.<br />
type ParseMonad = Either ParseError<br />
<br />
-- parseHexDigit attempts to convert a single hex digit into<br />
-- an Integer in the ParseMonad monad and throws an error on an<br />
-- invalid character<br />
parseHexDigit :: Char -> Int -> ParseMonad Integer<br />
parseHexDigit c idx = if isHexDigit c then<br />
return (toInteger (digitToInt c))<br />
else<br />
throwError (Err idx ("Invalid character '" ++ [c] ++ "'"))<br />
<br />
-- parseHex parses a string containing a hexadecimal number into<br />
-- an Integer in the ParseMonad monad. A parse error from parseHexDigit<br />
-- will cause an exceptional return from parseHex.<br />
parseHex :: String -> ParseMonad Integer<br />
parseHex s = parseHex' s 0 1<br />
where parseHex' [] val _ = return val<br />
parseHex' (c:cs) val idx = do d <- parseHexDigit c idx<br />
parseHex' cs ((val * 16) + d) (idx + 1)<br />
<br />
-- toString converts an Integer into a String in the ParseMonad monad<br />
toString :: Integer -> ParseMonad String<br />
toString n = return $ show n<br />
<br />
-- convert takes a String containing a hexadecimal representation of<br />
-- a number to a String containing a decimal representation of that<br />
-- number. A parse error on the input String will generate a<br />
-- descriptive error message as the output String.<br />
convert :: String -> String<br />
convert s = let (Right str) = do {n <- parseHex s; toString n} `catchError` printError<br />
in str<br />
where printError e = return $ "At index " ++ (show (location e)) ++ ":" ++ (reason e)<br />
</haskell><br />
The List monad<br />
<br />
= The List monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which may return 0, 1, or more possible results.<br />
<br />
Binding strategy:<br />
<br />
The bound function is applied to all possible values in the input list and the resulting lists are concatenated to produce a list of all possible results.<br />
<br />
Useful for:<br />
<br />
Building computations from sequences of non-deterministic operations. Parsing ambiguous grammars is a common example.<br />
<br />
Zero and plus:<br />
<br />
<code>[]</code> is the zero and <code>++</code> is the plus operation.<br />
<br />
Example type:<br />
<br />
<code>[a]</code><br />
<br />
== Motivation ==<br />
<br />
The List monad embodies the strategy of combining a chain of non-deterministic computations by applying the operations to all possible values at each step. It is useful when computations must deal with ambiguity. In that case it allows all possibilities to be explored until the ambiguity is resolved.<br />
<br />
== Definition ==<br />
<br />
<haskell><br />
instance Monad [] where<br />
m >>= f = concatMap f m<br />
return x = [x]<br />
fail s = []<br />
<br />
instance MonadPlus [] where<br />
mzero = []<br />
mplus = (++)<br />
</haskell><br />
== Example ==<br />
<br />
The canonical example of using the List monad is for parsing ambiguous grammars. The example below shows just a small example of parsing data into hex values, decimal values and words containing only alpha-numeric characters. Note that hexadecimal digits overlap both decimal digits and alphanumeric characters, leading to an ambiguous grammar. &quot;dead&quot; is both a valid hex value and a word, for example, and &quot;10&quot; is both a decimal value of 10 and a hex value of 16.<br />
<br />
Code available in [[../examples/example13.hs|example13.hs]]<br />
<br />
<haskell><br />
-- we can parse three different types of terms<br />
data Parsed = Digit Integer | Hex Integer | Word String deriving Show<br />
<br />
-- attempts to add a character to the parsed representation of a hex digit<br />
parseHexDigit :: Parsed -> Char -> [Parsed]<br />
parseHexDigit (Hex n) c = if isHexDigit c then<br />
return (Hex ((n*16) + (toInteger (digitToInt c))))<br />
else<br />
mzero<br />
parseHexDigit _ _ = mzero<br />
<br />
-- attempts to add a character to the parsed representation of a decimal digit<br />
parseDigit :: Parsed -> Char -> [Parsed]<br />
parseDigit (Digit n) c = if isDigit c then<br />
return (Digit ((n*10) + (toInteger (digitToInt c))))<br />
else<br />
mzero<br />
parseDigit _ _ = mzero<br />
<br />
-- attempts to add a character to the parsed representation of a word<br />
parseWord :: Parsed -> Char -> [Parsed]<br />
parseWord (Word s) c = if isAlpha c then<br />
return (Word (s ++ [c]))<br />
else<br />
mzero<br />
parseWord _ _ = mzero<br />
<br />
-- tries to parse the digit as a hex value, a decimal value and a word<br />
-- the result is a list of possible parses<br />
parse :: Parsed -> Char -> [Parsed]<br />
parse p c = (parseHexDigit p c) `mplus` (parseDigit p c) `mplus` (parseWord p c)<br />
<br />
-- parse an entire String and return a list of the possible parsed values<br />
parseArg :: String -> [Parsed]<br />
parseArg s = do init <- (return (Hex 0)) `mplus` (return (Digit 0)) `mplus` (return (Word ""))<br />
foldM parse init s<br />
</haskell><br />
The IO monad<br />
<br />
= The IO monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which perform I/O<br />
<br />
Binding strategy:<br />
<br />
I/O actions are executed in the order in which they are bound. Failures throw I/O errors which can be caught and handled.<br />
<br />
Useful for:<br />
<br />
Performing I/O within a Haskell program.<br />
<br />
Zero and plus:<br />
<br />
None.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/onlinelibrary/io.html IO a]<br />
<br />
== Motivation ==<br />
<br />
Input/Output is incompatible with a pure functional language because it is not referentially transparent and side-effect free. The IO monad solves this problem by confining computations that perform I/O within the IO monad.<br />
<br />
== Definition ==<br />
<br />
The definition of the IO monad is platform-specific. No data constructors are exported and no functions are provided to remove data from the IO monad. This makes the IO monad a one-way monad and is essential to ensuring safety of functional programs by isolating side-effects and non-referentially transparent actions within the imperative-style computations of the IO monad.<br />
<br />
Throughout this tutorial, we have referred to monadic values as computations. However, values in the IO monad are often called I/O actions and we will use that terminology here.<br />
<br />
In Haskell, the top-level <code>main</code> function must have type <code>IO ()</code>, so that programs are typically structured at the top level as an imperative-style sequence of I/O actions and calls to functional-style code. The functions exported from the <code>IO</code> module do not perform I/O themselves. They return I/O actions, which describe an I/O operation to be performed. The I/O actions are combined within the IO monad (in a purely functional manner) to create more complex I/O actions, resulting in the final I/O action that is the <code>main</code> value of the program.<br />
<br />
The standard prelude and the [http://www.haskell.org/onlinelibrary/io.html <code>IO</code> module] define many functions that can be used within the IO monad and any Haskell programmer will undoubtedly be familiar with some of them. This tutorial will only discuss the monadic aspects of the IO monad, not the full range of functions available to perform I/O.<br />
<br />
The <code>IO</code> type constructor is a member of the <code>Monad</code> class and the <code>MonadError</code> class, where errors are of the type <code>IOError</code>. <code>fail</code> is defined to throw an error built from the string argument. Within the <code>IO</code> monad you can use the exception mechanisms from the <code>Control.Monad.Error</code> module in the Monad Template Library if you import the module. The same mechanisms have alternative names exported by the <code>IO</code> module: <code>ioError</code> and <code>catch</code>.<br />
<br />
<haskell><br />
instance Monad IO where<br />
return a = ... -- function from a -> IO a<br />
m >>= k = ... -- executes the I/O action m and binds the value to k's input <br />
fail s = ioError (userError s)<br />
<br />
data IOError = ...<br />
<br />
ioError :: IOError -> IO a<br />
ioError = ...<br />
<br />
userError :: String -> IOError<br />
userError = ...<br />
<br />
catch :: IO a -> (IOError -> IO a) -> IO a <br />
catch = ...<br />
<br />
try :: IO a -> IO (Either IOError a)<br />
try f = catch (do r <- f<br />
return (Right r))<br />
(return . Left)<br />
</haskell><br />
The <code>IO</code> monad is incorporated into the Monad Template Library framework as an instance of the <code>MonadError</code> class.<br />
<br />
<haskell><br />
instance Error IOError where<br />
...<br />
<br />
instance MonadError IO where<br />
throwError = ioError<br />
catchError = catch<br />
</haskell><br />
The <code>IO</code> module exports a convenience function called <code>try</code> that executes an I/O action and returns <code>Right result</code> if the action succeeded or <code>Left IOError</code> if an I/O error was caught.<br />
<br />
== Example ==<br />
<br />
This example shows a partial implementation of the &quot;tr&quot; command that copies the standard input stream to the standard output stream with character translations controlled by command-line arguments. It demonstrates the use of the exception handling mechanisms of the <code>MonadError</code> class with the <code>IO</code> monad.<br />
<br />
Code available in [[../examples/example14.hs|example14.hs]]<br />
<br />
<haskell><br />
import Monad<br />
import System<br />
import IO<br />
import Control.Monad.Error<br />
<br />
-- translate char in set1 to corresponding char in set2<br />
translate :: String -> String -> Char -> Char<br />
translate [] _ c = c<br />
translate (x:xs) [] c = if x == c then ' ' else translate xs [] c<br />
translate (x:xs) [y] c = if x == c then y else translate xs [y] c<br />
translate (x:xs) (y:ys) c = if x == c then y else translate xs ys c<br />
<br />
-- translate an entire string<br />
translateString :: String -> String -> String -> String<br />
translateString set1 set2 str = map (translate set1 set2) str<br />
<br />
usage :: IOError -> IO ()<br />
usage e = do putStrLn "Usage: ex14 set1 set2"<br />
putStrLn "Translates characters in set1 on stdin to the corresponding"<br />
putStrLn "characters from set2 and writes the translation to stdout."<br />
<br />
-- translates stdin to stdout based on commandline arguments<br />
main :: IO ()<br />
main = (do [set1,set2] <- getArgs<br />
contents <- hGetContents stdin<br />
putStr $ translateString set1 set2 contents)<br />
`catchError` usage<br />
</haskell><br />
The State monad<br />
<br />
= The State monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which maintain state.<br />
<br />
Binding strategy:<br />
<br />
Binding threads a state parameter through the sequence of bound functions so that the same state value is never used twice, giving the illusion of in-place update.<br />
<br />
Useful for:<br />
<br />
Building computations from sequences of operations that require a shared state.<br />
<br />
Zero and plus:<br />
<br />
None.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.State.html State st a]<br />
<br />
== Motivation ==<br />
<br />
A pure functional language cannot update values in place because it violates referential transparency. A common idiom to simulate such stateful computations is to &quot;thread&quot; a state parameter through a sequence of functions:<br />
<br />
<haskell><br />
data MyType = MT Int Bool Char Int deriving Show<br />
<br />
makeRandomValue :: StdGen -> (MyType, StdGen)<br />
makeRandomValue g = let (n,g1) = randomR (1,100) g<br />
(b,g2) = random g1<br />
(c,g3) = randomR ('a','z') g2 <br />
(m,g4) = randomR (-n,n) g3<br />
in (MT n b c m, g4)<br />
</haskell><br />
This approach works, but such code can be error-prone, messy and difficult to maintain. The State monad hides the threading of the state parameter inside the binding operation, simultaneously making the code easier to write, easier to read and easier to modify.<br />
<br />
== Definition ==<br />
<br />
The definition shown here uses multi-parameter type classes and funDeps, which are not standard Haskell 98. It is not necessary to fully understand these details to make use of the State monad.<br />
<br />
<haskell><br />
newtype State s a = State { runState :: (s -> (a,s)) } <br />
<br />
instance Monad (State s) where <br />
return a = State $ \s -> (a,s)<br />
(State x) >>= f = State $ \s -> let (v,s') = x s in runState (f v) s' <br />
</haskell><br />
Values in the State monad are represented as transition functions from an initial state to a (value,newState) pair and a new type definition is provided to describe this construct: <code>State s a</code> is the type of a value of type <code>a</code> inside the State monad with state of type <code>s</code>.<br />
<br />
The type constructor <code>State s</code> is an instance of the <code>Monad</code> class. The <code>return</code> function simply creates a state transition function which sets the value but leaves the state unchanged. The binding operator creates a state transition function that applies its right-hand argument to the value and new state from its left-hand argument.<br />
<br />
<haskell><br />
class MonadState m s | m -> s where <br />
get :: m s<br />
put :: s -> m ()<br />
<br />
instance MonadState (State s) s where <br />
get = State $ \s -> (s,s) <br />
put s = State $ \_ -> ((),s) <br />
</haskell><br />
The <code>MonadState</code> class provides a standard but very simple interface for State monads. The <code>get</code> function retrieves the state by copying it as the value. The <code>put</code> function sets the state of the monad and does not yield a value.<br />
<br />
There are many additional functions provide which perform more complex computations built on top of <code>get</code> and put. The most useful one is <code>gets</code> which retrieves a function of the state. The others are listed in the [http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.State.html documentation] for the State monad library.<br />
<br />
== Example ==<br />
<br />
A simple application of the State monad is to thread the random generator state through multiple calls to the generation function.<br />
<br />
Code available in [[../examples/example15.hs|example15.hs]]<br />
<br />
<haskell><br />
data MyType = MT Int Bool Char Int deriving Show<br />
<br />
{- Using the State monad, we can define a function that returns<br />
a random value and updates the random generator state at<br />
the same time.<br />
-}<br />
getAny :: (Random a) => State StdGen a<br />
getAny = do g <- get<br />
(x,g') <- return $ random g<br />
put g'<br />
return x<br />
<br />
-- similar to getAny, but it bounds the random value returned<br />
getOne :: (Random a) => (a,a) -> State StdGen a<br />
getOne bounds = do g <- get<br />
(x,g') <- return $ randomR bounds g<br />
put g'<br />
return x<br />
<br />
{- Using the State monad with StdGen as the state, we can build<br />
random complex types without manually threading the<br />
random generator states through the code.<br />
-} <br />
makeRandomValueST :: StdGen -> (MyType, StdGen)<br />
makeRandomValueST = runState (do n <- getOne (1,100)<br />
b <- getAny<br />
c <- getOne ('a','z')<br />
m <- getOne (-n,n)<br />
return (MT n b c m))<br />
</haskell><br />
The Reader monad<br />
<br />
= The Reader monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which read values from a shared environment.<br />
<br />
Binding strategy:<br />
<br />
Monad values are functions from the environment to a value. The bound function is applied to the bound value, and both have access to the shared environment.<br />
<br />
Useful for:<br />
<br />
Maintaining variable bindings, or other shared environment.<br />
<br />
Zero and plus:<br />
<br />
None.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.Reader.html Reader [(String,Value)] a]<br />
<br />
== Motivation ==<br />
<br />
Some programming problems require computations within a shared environment (such as a set of variable bindings). These computations typically read values from the environment and sometimes execute sub-computations in a modified environment (with new or shadowing bindings, for example), but they do not require the full generality of the State monad.<br />
<br />
The Reader monad is specifically designed for these types of computations and is often a clearer and easier mechanism than using the State monad.<br />
<br />
== Definition ==<br />
<br />
The definition shown here uses multi-parameter type classes and funDeps, which are not standard Haskell 98. It is not necessary to fully understand these details to make use of the Reader monad.<br />
<br />
<haskell><br />
newtype Reader e a = Reader { runReader :: (e -> a) }<br />
<br />
instance Monad (Reader e) where <br />
return a = Reader $ \e -> a <br />
(Reader r) >>= f = Reader $ \e -> f (r e) e <br />
</haskell><br />
Values in the Reader monad are functions from an environment to a value. To extract the final value from a computation in the Reader monad, you simply apply <code>(runReader reader)</code> to an environment value.<br />
<br />
The <code>return</code> function creates a <code>Reader</code> that ignores the environment and produces the given value. The binding operator produces a <code>Reader</code> that uses the environment to extract the value its left-hand side and then applies the bound function to that value in the same environment.<br />
<br />
<haskell><br />
class MonadReader e m | m -> e where <br />
ask :: m e<br />
local :: (e -> e) -> m a -> m a <br />
<br />
instance MonadReader (Reader e) where <br />
ask = Reader id <br />
local f c = Reader $ \e -> runReader c (f e) <br />
<br />
asks :: (MonadReader e m) => (e -> a) -> m a <br />
asks sel = ask >>= return . sel<br />
</haskell><br />
The <code>MonadReader</code> class provides a number of convenience functions that are very useful when working with a Reader monad. The <code>ask</code> function retrieves the environment and the <code>local</code> function executes a computation in a modified environment. The <code>asks</code> function is a convenience function that retrieves a function of the current environment, and is typically used with a selector or lookup function.<br />
<br />
== Example ==<br />
<br />
Consider the problem of instantiating templates which contain variable substitutions and included templates. Using the Reader monad, we can maintain an environment of all known templates and all known variable bindings. Then, when a variable substitution is encountered, we can use the <code>asks</code> function to lookup the value of the variable. When a template is included with new variable definitions, we can use the <code>local</code> function to resolve the template in a modified environment that contains the additional variable bindings.<br />
<br />
Code available in [[../examples/example16.hs|example16.hs]]<br />
<br />
<haskell><br />
-- This the abstract syntax representation of a template<br />
-- Text Variable Quote Include Compound<br />
data Template = T String | V Template | Q Template | I Template [Definition] | C [Template]<br />
data Definition = D Template Template<br />
<br />
-- Our environment consists of an association list of named templates and<br />
-- an association list of named variable values. <br />
data Environment = Env {templates::[(String,Template)],<br />
variables::[(String,String)]}<br />
<br />
-- lookup a variable from the environment<br />
lookupVar :: String -> Environment -> Maybe String<br />
lookupVar name env = lookup name (variables env)<br />
<br />
-- lookup a template from the environment<br />
lookupTemplate :: String -> Environment -> Maybe Template<br />
lookupTemplate name env = lookup name (templates env)<br />
<br />
-- add a list of resolved definitions to the environment<br />
addDefs :: [(String,String)] -> Environment -> Environment<br />
addDefs defs env = env {variables = defs ++ (variables env)}<br />
<br />
-- resolve a Definition and produce a (name,value) pair<br />
resolveDef :: Definition -> Reader Environment (String,String)<br />
resolveDef (D t d) = do name <- resolve t<br />
value <- resolve d<br />
return (name,value)<br />
<br />
-- resolve a template into a string<br />
resolve :: Template -> Reader Environment (String)<br />
resolve (T s) = return s<br />
resolve (V t) = do varName <- resolve t<br />
varValue <- asks (lookupVar varName)<br />
return $ maybe "" id varValue<br />
resolve (Q t) = do tmplName <- resolve t<br />
body <- asks (lookupTemplate tmplName)<br />
return $ maybe "" show body <br />
resolve (I t ds) = do tmplName <- resolve t<br />
body <- asks (lookupTemplate tmplName)<br />
case body of<br />
Just t' -> do defs <- mapM resolveDef ds<br />
local (addDefs defs) (resolve t')<br />
Nothing -> return ""<br />
resolve (C ts) = (liftM concat) (mapM resolve ts)<br />
</haskell><br />
To use the Reader monad to resolve a template <code>t</code> into a <code>String</code>, you simply need to do <code>runReader (resolve t) env</code>.<br />
<br />
The Writer monad<br />
<br />
= The Writer monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which produce a stream of data in addition to the computed values.<br />
<br />
Binding strategy:<br />
<br />
A Writer monad value is a (computation value, log value) pair. Binding replaces the computation value with the result of applying the bound function to the previous value and appends any log data from the computation to the existing log data.<br />
<br />
Useful for:<br />
<br />
Logging, or other computations that produce output &quot;on the side&quot;.<br />
<br />
Zero and plus:<br />
<br />
None.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.Writer.html Writer [String] a]<br />
<br />
== Motivation ==<br />
<br />
It is often desirable for a computation to generate output &quot;on the side&quot;. Logging and tracing are the most common examples in which data is generated during a computation that we want to retain but is not the primary result of the computation.<br />
<br />
Explicitly managing the logging or tracing data can clutter up the code and invite subtle bugs such as missed log entries. The Writer monad provides a cleaner way to manage the output without cluttering the main computation.<br />
<br />
== Definition ==<br />
<br />
The definition shown here uses multi-parameter type classes and funDeps, which are not standard Haskell 98. It is not necessary to fully understand these details to make use of the Writer monad.<br />
<br />
[[Image:info.png]] To fully understand this definition, you need to know about Haskell's <code>Monoid</code> class, which represents a mathematical structure called a monoid in the same way that the <code>Monad</code> class represents the monad structure.<br />
<br />
The good news is that monoids are simpler than monads. A monoid is a set of objects, a single identity element, and an associative binary operator over the set of objects. A monoid must obey some mathematical laws, such that applying the operator to any values from the set gives another value in the set, and whenever one operand of the operator is the identity element the result is equal to the other operand. You may notice that these laws are the same as the laws governing <code>mzero</code> and <code>mplus</code> for instances of <code>MonadPlus</code>. That is because monads with a zero and plus are monads that are also monoids!<br />
<br />
Some examples of mathematical monoids are the natural numbers with identity element 0 and binary operator for addition, and also the natural numbers with identity element 1 and binary operator for multiplication.<br />
<br />
In Haskell, a monoid consists of a type, an identity element, and a binary operator. Haskell defines the <code>Monoid</code> class (in Data.Monoid) to provide a standard convention for working with monoids: the identity element is named <code>mempty</code> and the operator is named <code>mappend</code>.<br />
<br />
The most commonly used standard monoid in Haskell is the list, but functions of type <code>(a -> a)</code> also form a monoid.<br />
<br />
[[Image:warn.png]] Care should be taken when using a list as the monoid for a Writer, as there may be a performance penalty associated with the <code>mappend</code> operation as the output grows. In that case, a data structure that supports fast append operations would be a more appropriate choice.<br />
<br />
<haskell><br />
newtype Writer w a = Writer { runWriter :: (a,w) } <br />
<br />
instance (Monoid w) => Monad (Writer w) where <br />
return a = Writer (a,mempty) <br />
(Writer (a,w)) >>= f = let (a',w') = runWriter $ f a in Writer (a',w `mappend` w')<br />
</haskell><br />
The Writer monad maintains a (value,log) pair, where the log type must be a monoid. The <code>return</code> function simply returns the value along with an empty log. Binding executes the bound function using the current value as input, and appends any log output to the existing log.<br />
<br />
<haskell><br />
class (Monoid w, Monad m) => MonadWriter w m | m -> w where <br />
pass :: m (a,w -> w) -> m a <br />
listen :: m a -> m (a,w) <br />
tell :: w -> m () <br />
<br />
instance (Monoid w) => MonadWriter (Writer w) where <br />
pass (Writer ((a,f),w)) = Writer (a,f w) <br />
listen (Writer (a,w)) = Writer ((a,w),w) <br />
tell s = Writer ((),s) <br />
<br />
listens :: (MonadWriter w m) => (w -> w) -> m a -> m (a,w) <br />
listens f m = do (a,w) <- m; return (a,f w)<br />
<br />
censor :: (MonadWriter w m) => (w -> w) -> m a -> m a <br />
censor f m = pass $ do a <- m; return (a,f)<br />
</haskell><br />
The <code>MonadWriter</code> class provides a number of convenience functions for working with Writer monads. The simplest and most useful is <code>tell</code>, which adds one or more entries to the log. The <code>listen</code> function turns a Writer that returns a value <code>a</code> and produces output <code>w</code> into a Writer that produces a value <code>(a,w)</code> and still produces output <code>w</code>. This allows the computation to &quot;listen&quot; to the log output generated by a Writer.<br />
<br />
The <code>pass</code> function is slightly more complicated. It converts a Writer that produces a value <code>(a,f)</code> and output <code>w</code> into a Writer that produces a value <code>a</code> and output <code>f w</code>. This is somewhat cumbersome, so the helper function <code>censor</code> is normally used. The <code>censor</code> function takes a function and a Writer and produces a new Writer whose output is the same but whose log entry has been modified by the function.<br />
<br />
The <code>listens</code> function operates just like <code>listen</code> except that the log part of the value is modified by the supplied function.<br />
<br />
== Example ==<br />
<br />
In this example, we imagine a very simple firewall that filters packets based on a rulebase of rules matching the source and destination hosts and the payload of the packet. The firewall's primary job is packet filtering, but we would also like it to produce a log of its activity.<br />
<br />
Code available in [[../examples/example17.hs|example17.hs]]<br />
<br />
<haskell><br />
-- this is the format of our log entries<br />
data Entry = Log {count::Int, msg::String} deriving Eq<br />
<br />
-- add a message to the log<br />
logMsg :: String -> Writer [Entry] ()<br />
logMsg s = tell [Log 1 s]<br />
<br />
-- this handles one packet<br />
filterOne :: [Rule] -> Packet -> Writer [Entry] (Maybe Packet)<br />
filterOne rules packet = do <br />
rule <- return (match rules packet)<br />
case rule of<br />
Nothing -> do<br />
logMsg $ "DROPPING UNMATCHED PACKET: " ++ (show packet)<br />
return Nothing<br />
(Just r) -> do<br />
when (logIt r) $ logMsg ("MATCH: " ++ (show r) ++ " <=> " ++ (show packet))<br />
case r of (Rule Accept _ _) -> return $ Just packet<br />
(Rule Reject _ _) -> return Nothing<br />
</haskell><br />
That was pretty simple, but what if we want to merge duplicate consecutive log entries? None of the existing functions allow us to modify the output from previous stages of the computation, but we can use a &quot;delayed logging&quot; trick to only add a log entry only after we get a new entry that doesn't match the ones before it.<br />
<br />
Code available in [[../examples/example17.hs|example17.hs]]<br />
<br />
<haskell><br />
-- merge identical entries at the end of the log<br />
-- This function uses [Entry] as both the log type and the result type.<br />
-- When two identical messages are merged, the result is just the message<br />
-- with an incremented count. When two different messages are merged,<br />
-- the first message is logged and the second is returned as the result.<br />
mergeEntries :: [Entry] -> [Entry] -> Writer [Entry] [Entry]<br />
mergeEntries [] x = return x<br />
mergeEntries x [] = return x<br />
mergeEntries [e1] [e2] = let (Log n msg) = e1<br />
(Log n' msg') = e2<br />
in if msg == msg' then<br />
return [(Log (n+n') msg)]<br />
else<br />
do tell [e1]<br />
return [e2]<br />
<br />
-- This is a complex-looking function but it is actually pretty simple.<br />
-- It maps a function over a list of values to get a list of Writers,<br />
-- then runs each writer and combines the results. The result of the function<br />
-- is a writer whose value is a list of all the values from the writers and whose<br />
-- log output is the result of folding the merge operator into the individual<br />
-- log entries (using 'initial' as the initial log value).<br />
groupSame :: (Monoid a) => a -> (a -> a -> Writer a a) -> [b] -> (b -> Writer a c) -> Writer a [c]<br />
groupSame initial merge [] _ = do tell initial<br />
return []<br />
groupSame initial merge (x:xs) fn = do (result,output) <- return (runWriter (fn x))<br />
new <- merge initial output<br />
rest <- groupSame new merge xs fn<br />
return (result:rest)<br />
<br />
-- this filters a list of packets, producing a filtered packet list and a log of<br />
-- the activity in which consecutive messages are merged<br />
filterAll :: [Rule] -> [Packet] -> Writer [Entry] [Packet]<br />
filterAll rules packets = do tell [Log 1 "STARTING PACKET FILTER"]<br />
out <- groupSame [] mergeEntries packets (filterOne rules)<br />
tell [Log 1 "STOPPING PACKET FILTER"]<br />
return (catMaybes out)<br />
</haskell><br />
The Continuation monad<br />
<br />
= The Continuation monad =<br />
<br />
== Overview ==<br />
<br />
Computation type:<br />
<br />
Computations which can be interrupted and resumed.<br />
<br />
Binding strategy:<br />
<br />
Binding a function to a monadic value creates a new continuation which uses the function as the continuation of the monadic computation.<br />
<br />
Useful for:<br />
<br />
Complex control structures, error handling and creating co-routines.<br />
<br />
Zero and plus:<br />
<br />
None.<br />
<br />
Example type:<br />
<br />
[http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.Cont.html Cont r a]<br />
<br />
== Motivation ==<br />
<br />
[[Image:warn.png]] Abuse of the Continuation monad can produce code that is impossible to understand and maintain.<br />
<br />
Before using the Continuation monad, be sure that you have a firm understanding of continuation-passing style (CPS) and that continuations represent the best solution to your particular design problem. Many algorithms which require continuations in other languages do not require them in Haskell, due to Haskell's lazy semantics.<br />
<br />
Continuations represent the ''future'' of a computation, as a function from an intermediate result to the final result. In continuation-passing style, computations are built up from sequences of nested continuations, terminated by a final continuation (often <code>id</code>) which produces the final result. Since continuations are functions which represent the future of a computation, manipulation of the continuation functions can achieve complex manipulations of the future of the computation, such as interrupting a computation in the middle, aborting a portion of a computation, restarting a computation and interleaving execution of computations. The Continuation monad adapts CPS to the structure of a monad.<br />
<br />
== Definition ==<br />
<br />
<haskell><br />
newtype Cont r a = Cont { runCont :: ((a -> r) -> r) } -- r is the final result type of the whole computation <br />
<br />
instance Monad (Cont r) where <br />
return a = Cont $ \k -> k a -- i.e. return a = \k -> k a <br />
(Cont c) >>= f = Cont $ \k -> c (\a -> runCont (f a) k) -- i.e. c >>= f = \k -> c (\a -> f a k) <br />
</haskell><br />
The Continuation monad represents computations in continuation-passing style. <code>Cont r a</code> is a CPS computation that produces an intermediate result of type <code>a</code> within a CPS computation whose final result type is <code>r</code>.<br />
<br />
The <code>return</code> function simply creates a continuation which passes the value on. The <code>>>=</code> operator adds the bound function into the continuation chain.<br />
<br />
<haskell><br />
class (Monad m) => MonadCont m where <br />
callCC :: ((a -> m b) -> m a) -> m a <br />
<br />
instance MonadCont (Cont r) where <br />
callCC f = Cont $ \k -> runCont (f (\a -> Cont $ \_ -> k a)) k<br />
</haskell><br />
The <code>MonadCont</code> class provides the <code>callCC</code> function, which provides an escape continuation mechanism for use with Continuation monads. Escape continuations allow you to abort the current computation and return a value immediately. They achieve a similar effect to <code>throwError</code> and catchError within an <code>Error</code> monad.<br />
<br />
<code>callCC</code> calls a function with the current continuation as its argument (hence the name). The standard idiom used with <code>callCC</code> is to provide a lambda-expression to name the continuation. Then calling the named continuation anywhere within its scope will escape from the computation, even if it is many layers deep within nested computations.<br />
<br />
In addition to the escape mechanism provided by <code>callCC</code>, the Continuation monad can be used to implement other, more powerful continuation manipulations. These other mechanisms have fairly specialized uses, however — and abuse of them can easily create fiendishly obfuscated code — so they will not be covered here.<br />
<br />
== Example ==<br />
<br />
This example gives a taste of how escape continuations work. The example function uses escape continuations to perform a complicated transformation on an integer.<br />
<br />
Code available in [[../examples/example18.hs|example18.hs]]<br />
<br />
<haskell><br />
{- We use the continuation monad to perform "escapes" from code blocks.<br />
This function implements a complicated control structure to process<br />
numbers:<br />
<br />
Input (n) Output List Shown<br />
========= ====== ==========<br />
0-9 n none<br />
10-199 number of digits in (n/2) digits of (n/2)<br />
200-19999 n digits of (n/2)<br />
20000-1999999 (n/2) backwards none<br />
>= 2000000 sum of digits of (n/2) digits of (n/2)<br />
-} <br />
fun :: Int -> String<br />
fun n = (`runCont` id) $ do<br />
str <- callCC $ \exit1 -> do -- define "exit1"<br />
when (n < 10) (exit1 (show n))<br />
let ns = map digitToInt (show (n `div` 2))<br />
n' <- callCC $ \exit2 -> do -- define "exit2"<br />
when ((length ns) < 3) (exit2 (length ns))<br />
when ((length ns) < 5) (exit2 n)<br />
when ((length ns) < 7) $ do let ns' = map intToDigit (reverse ns)<br />
exit1 (dropWhile (=='0') ns') --escape 2 levels<br />
return $ sum ns<br />
return $ "(ns = " ++ (show ns) ++ ") " ++ (show n')<br />
return $ "Answer: " ++ str<br />
</haskell><br />
Part III - Introduction<br />
<br />
= Introduction =<br />
<br />
Part I has introduced the monad concept and Part II has provided an understanding of a number of common, useful monads in the standard Haskell libraries. This is not enough to put monads into heavy practice, however, because in the real world you often want computations which combine aspects of more than one monad at the same time, such as stateful, non-determistic computations or computations which make use of continuations and perform I/O. When one computation is a strict subset of the other, it is possible to perform the monad computations separately, unless the sub-computation is performed in a one-way monad.<br />
<br />
Often, the computations can't be performed in isolation. In this case, what is needed is a monad that combines the features of the two monads into a single computation. It is inefficient and poor practice to write a new monad instance with the required characteristics each time a new combination is desired. Instead, we would prefer to develop a way to combine the standard monads to produce the needed hybrids. The technique that lets us do exactly that is called monad transformers.<br />
<br />
Monad transformers are the topic of Part III, and they are explained by revisiting earlier examples to see how monad transformers can be used to add more realistic capabilities to them. It may be helpful to review the earlier examples as they are re-examined.<br />
<br />
Combining monads the hard way<br />
<br />
= Combining monads the hard way =<br />
<br />
Before we investigate the use of monad transformers, we will see how monads can be combined without using transformers. This is a useful excercise to develop insights into the issues that arise when combining monads and provides a baseline from which the advantages of the transformer approach can be measured. We use the code from [[contmonad.html#example|example 18]] (the Continuation monad) to illustrate these issues, so you may want to review it before continuing.<br />
<br />
== Nested Monads ==<br />
<br />
Some computations have a simple enough structure that the monadic computations can be nested, avoiding the need for a combined monad altogether. In Haskell, all computations occur in the IO monad at the top level, so the monad examples we have seen so far all actually use the technique of nested monadic computations. To do this, the computations perform all of their input at the beginning — usually by reading arguments from the command line — then pass the values on to the monadic computations to produce results, and finally perform their output at the end. This structure avoids the issues of combining monads but makes the examples seem contrived at times.<br />
<br />
The code introduced in example 18 followed the nesting pattern: reading a number from the command line in the IO monad, passing that number to a computation in the Continuation monad to produce a string, and then writing the string back in the IO monad. The computations in the IO monad aren't restricted to reading from the command line and writing strings; they can be arbitrarily complex. Likewise, the inner computation can be arbitrarily complex as well. As long as the inner computation does not depend on the functionality of the outer monad, it can be safely nested within the outer monad, as illustrated in this variation on example 18 which reads the value from stdin instead of using a command line argument:<br />
<br />
Code available in [[../examples/example19.hs|example19.hs]]<br />
<br />
<haskell><br />
fun :: IO String<br />
fun = do n <- (readLn::IO Int) -- this is an IO monad block<br />
return $ (`runCont` id) $ do -- this is a Cont monad block<br />
str <- callCC $ \exit1 -> do<br />
when (n < 10) (exit1 (show n))<br />
let ns = map digitToInt (show (n `div` 2))<br />
n' <- callCC $ \exit2 -> do<br />
when ((length ns) < 3) (exit2 (length ns))<br />
when ((length ns) < 5) (exit2 n)<br />
when ((length ns) < 7) $ do let ns' = map intToDigit (reverse ns)<br />
exit1 (dropWhile (=='0') ns')<br />
return $ sum ns<br />
return $ "(ns = " ++ (show ns) ++ ") " ++ (show n')<br />
return $ "Answer: " ++ str<br />
</haskell><br />
== Combined Monads ==<br />
<br />
What about computations with more complicated structure? If the nesting pattern cannot be used, we need a way to combine the attributes of two or more monads in a single computation. This is accomplished by doing computations within a monad in which the values are themselves monadic values in another monad. For example, we might perform computations in the Continuation monad of type <code>Cont (IO String) a</code> if we need to perform I/O within the computation in the Continuation monad. We could use a monad of type <code>State (Either Err a) a</code> to combine the features of the State and Error monads in a single computation.<br />
<br />
Consider a slight modification to our example in which we perform the same I/O at the beginning, but we may require additional input in the middle of the computation in the Continuation monad. In this case, we will allow the user to specify part of the output value when the input value is within a certain range. Because the I/O depends on part of the computation in the Continuation monad and part of the computation in the Continuation monad depends on the result of the I/O, we cannot use the nested monad pattern.<br />
<br />
Instead, we make the computation in the Continuation monad use values from the IO monad. What used to be <code>Int</code> and <code>String</code> values are now of type <code>IO Int</code> and <code>IO String</code>. We can't extract values from the IO monad — it's a one-way monad — so we may need to nest little do-blocks of the IO monad within the Continuation monad to manipulate the values. We use a helper function <code>toIO</code> to make it clearer when we are creating values in the IO monad nested within the Continuation monad.<br />
<br />
Code available in [[../examples/example20.hs|example20.hs]]<br />
<br />
<haskell><br />
toIO :: a -> IO a<br />
toIO x = return x<br />
<br />
fun :: IO String<br />
fun = do n <- (readLn::IO Int) -- this is an IO monad block<br />
convert n<br />
<br />
convert :: Int -> IO String<br />
convert n = (`runCont` id) $ do -- this is a Cont monad block<br />
str <- callCC $ \exit1 -> do -- str has type IO String<br />
when (n < 10) (exit1 $ toIO (show n))<br />
let ns = map digitToInt (show (n `div` 2))<br />
n' <- callCC $ \exit2 -> do -- n' has type IO Int<br />
when ((length ns) < 3) (exit2 (toIO (length ns)))<br />
when ((length ns) < 5) (exit2 $ do putStrLn "Enter a number:"<br />
x <- (readLn::IO Int)<br />
return x)<br />
when ((length ns) < 7) $ do let ns' = map intToDigit (reverse ns)<br />
exit1 $ toIO (dropWhile (=='0') ns')<br />
return (toIO (sum ns))<br />
return $ do num <- n' -- this is an IO monad block<br />
return $ "(ns = " ++ (show ns) ++ ") " ++ (show num)<br />
return $ do s <- str -- this is an IO monad block<br />
return $ "Answer: " ++ s<br />
</haskell><br />
Even this trivial example has gotten confusing and ugly when we tried to combine different monads in the same computation. It works, but it isn't pretty. Comparing the code side-by-side shows the degree to which the manual monad combination strategy pollutes the code.<br />
<br />
Nested monads from example 19<br />
<br />
Manually combined monads from example 20<br />
<br />
<haskell><br />
fun = do n <- (readLn::IO Int)<br />
return $ (`runCont` id) $ do<br />
str <- callCC $ \exit1 -> do<br />
when (n < 10) (exit1 (show n))<br />
let ns = map digitToInt (show (n `div` 2))<br />
n' <- callCC $ \exit2 -> do<br />
when ((length ns) < 3) (exit2 (length ns))<br />
when ((length ns) < 5) (exit2 n)<br />
when ((length ns) < 7) $ do<br />
let ns' = map intToDigit (reverse ns)<br />
exit1 (dropWhile (=='0') ns')<br />
return $ sum ns<br />
return $ "(ns = " ++ (show ns) ++ ") " ++ (show n')<br />
return $ "Answer: " ++ str<br />
</haskell><br />
<haskell><br />
convert n = (`runCont` id) $ do<br />
str <- callCC $ \exit1 -> do<br />
when (n < 10) (exit1 $ toIO (show n))<br />
let ns = map digitToInt (show (n `div` 2))<br />
n' <- callCC $ \exit2 -> do<br />
when ((length ns) < 3) (exit2 (toIO (length ns)))<br />
when ((length ns) < 5) (exit2 $ do<br />
putStrLn "Enter a number:"<br />
x <- (readLn::IO Int)<br />
return x)<br />
when ((length ns) < 7) $ do<br />
let ns' = map intToDigit (reverse ns)<br />
exit1 $ toIO (dropWhile (=='0') ns')<br />
return (toIO (sum ns))<br />
return $ do num <- n'<br />
return $ "(ns = " ++ (show ns) ++ ") " ++ (show num)<br />
return $ do s <- str<br />
return $ "Answer: " ++ s<br />
</haskell><br />
Monad transformers<br />
<br />
= Monad transformers =<br />
<br />
Monad transformers are special variants of standard monads that facilitate the combining of monads. Their type constructors are parameterized over a monad type constructor, and they produce combined monadic types.<br />
<br />
== Transformer type constructors ==<br />
<br />
Type constructors play a fundamental role in Haskell's monad support. Recall that <code>Reader r a</code> is the type of values of type <code>a</code> within a Reader monad with environment of type <code>r</code>. The type const