Monad
From HaskellWiki
m (fixed link to Identity Monad) |
m (→Commutative monads: no reason to restrict this to functions that return actions) |
||
(24 intermediate revisions by 5 users not shown) | |||
Line 1: | Line 1: | ||
{{Standard class|Monad|module=Control.Monad|module-doc=Control-Monad|package=base}} | {{Standard class|Monad|module=Control.Monad|module-doc=Control-Monad|package=base}} | ||
− | '''''Monads''''' in Haskell can be thought of as ''composable'' computation descriptions. The essence of monad is thus ''separation'' of ''composition timeline'' from the composed computation's ''execution timeline'', as well as the ability of ''computation'' to implicitly carry extra data as pertaining to the computation itself in addition to its ''one'' (hence the name) output. This lends monads to supplementing ''pure'' calculations with features like I/O, common environment or state, and to ''preprocessing'' of computations (simplification, optimization etc.). | + | '''''Monads''''' in Haskell can be thought of as ''composable'' computation descriptions. The essence of monad is thus ''separation'' of ''composition timeline'' from the composed computation's ''execution timeline'', as well as the ability of ''computation'' to implicitly carry extra data, as pertaining to the computation itself, in addition to its ''one'' (hence the name) output, that it '''''will produce''''' when run (or queried, or called upon). This lends monads to supplementing ''pure'' calculations with features like I/O, common environment or state, and to ''preprocessing'' of computations (simplification, optimization etc.). |
− | Each monad, or computation type, provides means, subject to '''''Monad Laws''''', | + | Each monad, or computation type, provides means, subject to '''''Monad Laws''''', to '''''(a)''''' ''create'' a description of computation action that will produce (a.k.a. "return") a given Haskell value, '''''(b)''''' somehow ''run'' a computation action description (possibly getting its output back into Haskell should the monad choose to allow it, if computations described by the monad are pure, or causing the prescribed side effects if it's not), and '''''(c)''''' ''combine'' (a.k.a. "bind") a computation action description with a ''reaction'' to it – a regular Haskell function of one argument (that will receive computation-produced value) returning another action description (using or dependent on that value, if need be) – thus creating a combined computation action description that will feed the original action's output through the reaction while automatically taking care of the particulars of the computational process itself. A monad might also define additional primitives to provide access to and/or enable manipulation of data it implicitly carries, specific to its nature. |
− | + | [[Image:Monads inter-dependencies 2.png|center]] | |
− | Because they are very useful in practice but rather mind-twisting for the beginners, numerous tutorials that deal exclusively with monads were created (see [[Monad#Monad tutorials|monad tutorials]]). | + | Thus in Haskell, though it is a purely-functional language, side effects that '''''will be performed''''' by a computation can be dealt with and combined ''purely'' at the monad's composition time. Monads thus resemble programs in a particular [[DSL]]. While programs may describe impure effects and actions ''outside'' Haskell, they can still be combined and processed (''"assembled"'') purely, ''inside'' Haskell, creating a pure Haskell value - a computation action description that describes an impure calculation. That is how Monads in Haskell '''''separate''''' between the ''pure'' and the ''impure''. |
+ | |||
+ | The computation doesn't have to be impure and can be pure itself as well. Then monads serve to provide the benefits of separation of concerns, and automatic creation of a computational "pipeline". Because they are very useful in practice but rather mind-twisting for the beginners, numerous tutorials that deal exclusively with monads were created (see [[Monad#Monad tutorials|monad tutorials]]). | ||
== Common monads == | == Common monads == | ||
Most common applications of monads include: | Most common applications of monads include: | ||
* Representing failure using <hask>Maybe</hask> monad | * Representing failure using <hask>Maybe</hask> monad | ||
− | * Nondeterminism | + | * Nondeterminism using <hask>List</hask> monad to represent carrying multiple values |
* State using <hask>State</hask> monad | * State using <hask>State</hask> monad | ||
* Read-only environment using <hask>Reader</hask> monad | * Read-only environment using <hask>Reader</hask> monad | ||
Line 37: | Line 39: | ||
</haskell> | </haskell> | ||
− | See [[Monad laws|this intuitive explanation]] of why they should obey the Monad laws. | + | See [[Monad laws|this intuitive explanation]] of why they should obey the Monad laws. It basically says that monad's reactions should be associative under Kleisli composition, defined as <code>(f >=> g) x = f x >>= g</code>, with <code>return</code> its left and right identity element. |
Any Monad can be made a [[Functor]] by defining | Any Monad can be made a [[Functor]] by defining | ||
Line 51: | Line 53: | ||
In order to improve the look of code that uses monads Haskell provides a special [[syntactic sugar]] called <hask>do</hask>-notation. For example, following expression: | In order to improve the look of code that uses monads Haskell provides a special [[syntactic sugar]] called <hask>do</hask>-notation. For example, following expression: | ||
<haskell> | <haskell> | ||
− | thing1 >>= (\x -> func1 x >>= (\y -> thing2 >>= (\_ -> func2 y (\z -> return z)))) | + | thing1 >>= (\x -> func1 x >>= (\y -> thing2 |
+ | >>= (\_ -> func2 y (\z -> return z)))) | ||
</haskell> | </haskell> | ||
which can be written more clearly by breaking it into several lines and omitting parentheses: | which can be written more clearly by breaking it into several lines and omitting parentheses: | ||
Line 82: | Line 85: | ||
<haskell> | <haskell> | ||
do | do | ||
− | a <- | + | a <- actA |
− | b <- | + | b <- actB |
m a b | m a b | ||
</haskell> | </haskell> | ||
Line 89: | Line 92: | ||
<haskell> | <haskell> | ||
do | do | ||
− | b <- | + | b <- actB |
− | a <- | + | a <- actA |
m a b | m a b | ||
</haskell> | </haskell> | ||
Line 102: | Line 105: | ||
Monads are known for being deeply confusing to lots of people, so there are plenty of tutorials specifically related to monads. Each takes a different approach to Monads, and hopefully everyone will find something useful. | Monads are known for being deeply confusing to lots of people, so there are plenty of tutorials specifically related to monads. Each takes a different approach to Monads, and hopefully everyone will find something useful. | ||
− | See [[ | + | See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials. |
== Monad reference guides == | == Monad reference guides == | ||
Line 119: | Line 122: | ||
* [http://www-static.cc.gatech.edu/~yannis/fc++/FC++.1.5/monad.h C++], [http://www-static.cc.gatech.edu/~yannis/fc++/New1.5/lambda.html#monad doc] | * [http://www-static.cc.gatech.edu/~yannis/fc++/FC++.1.5/monad.h C++], [http://www-static.cc.gatech.edu/~yannis/fc++/New1.5/lambda.html#monad doc] | ||
* [http://cml.cs.uchicago.edu/pages/cml.html CML.event] ? | * [http://cml.cs.uchicago.edu/pages/cml.html CML.event] ? | ||
− | * [http:// | + | * [http://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf Clean] State monad |
* [http://clojure.googlegroups.com/web/monads.clj Clojure] | * [http://clojure.googlegroups.com/web/monads.clj Clojure] | ||
* [http://cratylus.freewebspace.com/monads-in-javascript.htm JavaScript] | * [http://cratylus.freewebspace.com/monads-in-javascript.htm JavaScript] | ||
− | * [http://www.ccs.neu.edu/home/dherman/code/monads/JavaMonads | + | * [http://www.ccs.neu.edu/home/dherman/browse/code/monads/JavaMonads/ Java] |
* [http://permalink.gmane.org/gmane.comp.lang.concatenative/1506 Joy] | * [http://permalink.gmane.org/gmane.comp.lang.concatenative/1506 Joy] | ||
− | * [http://research.microsoft.com/~emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ], [http://www.idealliance.org/xmlusa/05/call/xmlpapers/63.1015/.63.html#S4. more, C#, VB] | + | * [http://research.microsoft.com/~emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ], [http://www.idealliance.org/xmlusa/05/call/xmlpapers/63.1015/.63.html#S4. more, C#, VB] (inaccessible) |
* [http://sleepingsquirrel.org/monads/monads.lisp Lisp] | * [http://sleepingsquirrel.org/monads/monads.lisp Lisp] | ||
* [http://lambda-the-ultimate.org/node/1136#comment-12448 Miranda] | * [http://lambda-the-ultimate.org/node/1136#comment-12448 Miranda] | ||
Line 132: | Line 135: | ||
** [http://www.pps.jussieu.fr/~beffara/darcs/pivm/caml-vm/monad.mli also] | ** [http://www.pps.jussieu.fr/~beffara/darcs/pivm/caml-vm/monad.mli also] | ||
** [http://www.cas.mcmaster.ca/~carette/metamonads/ MetaOcaml] | ** [http://www.cas.mcmaster.ca/~carette/metamonads/ MetaOcaml] | ||
− | ** [http://enfranchisedmind.com/blog/ | + | ** [http://enfranchisedmind.com/blog/posts/a-monad-tutorial-for-ocaml/ A Monad Tutorial for Ocaml] |
* [http://sleepingsquirrel.org/monads/monads.html Perl] | * [http://sleepingsquirrel.org/monads/monads.html Perl] | ||
* [http://programming.reddit.com/info/p66e/comments Perl6 ?] | * [http://programming.reddit.com/info/p66e/comments Perl6 ?] | ||
Line 178: | Line 181: | ||
* [http://haskell.org/haskellwiki/New_monads/MonadSupply Unique supply] | * [http://haskell.org/haskellwiki/New_monads/MonadSupply Unique supply] | ||
* [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-ST.html ST - memory-only effects] | * [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-ST.html ST - memory-only effects] | ||
− | * [http://haskell.org/ | + | * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-State.html Global state] |
* [http://haskell.org/haskellwiki/New_monads/MonadUndo Undoable state effects] | * [http://haskell.org/haskellwiki/New_monads/MonadUndo Undoable state effects] | ||
* [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-Instances.html Function application] | * [http://haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-Instances.html Function application] | ||
− | * [http://haskell.org/ | + | * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Error.html Functions which may error] |
* [http://haskell.org/ghc/docs/latest/html/libraries/stm/Control-Monad-STM.html Atomic memory transactions] | * [http://haskell.org/ghc/docs/latest/html/libraries/stm/Control-Monad-STM.html Atomic memory transactions] | ||
− | * [http://haskell.org/ | + | * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Cont.html Continuations] |
* [http://haskell.org/ghc/docs/latest/html/libraries/base/System-IO.html#t%3AIO IO - unrestricted side effects] | * [http://haskell.org/ghc/docs/latest/html/libraries/base/System-IO.html#t%3AIO IO - unrestricted side effects] | ||
* [http://www.haskell.org/haskellwiki/Sudoku Non-deterministic evaluation] | * [http://www.haskell.org/haskellwiki/Sudoku Non-deterministic evaluation] | ||
Line 212: | Line 215: | ||
* If you are tired of monads, you can easily [http://saxophone.jpberlin.de/MonadTransformer?source=http%3A%2F%2Fwww%2Ehaskell%2Eorg%2Fhaskellwiki%2FCategory%3AMonad&language=English get rid of them]. | * If you are tired of monads, you can easily [http://saxophone.jpberlin.de/MonadTransformer?source=http%3A%2F%2Fwww%2Ehaskell%2Eorg%2Fhaskellwiki%2FCategory%3AMonad&language=English get rid of them]. | ||
+ | |||
+ | ==See also== | ||
+ | |||
+ | * [[What a Monad is not]] | ||
+ | * [[Monads as containers]] | ||
+ | * [[Monads as computation]] | ||
+ | * [[Monad/ST]] | ||
+ | * [http://www.haskellforall.com/2012/06/you-could-have-invented-free-monads.html Why free monads matter] (blog article) | ||
[[Category:Monad|*]] | [[Category:Monad|*]] |
Revision as of 22:09, 3 September 2012
import Control.Monad |
Monads in Haskell can be thought of as composable computation descriptions. The essence of monad is thus separation of composition timeline from the composed computation's execution timeline, as well as the ability of computation to implicitly carry extra data, as pertaining to the computation itself, in addition to its one (hence the name) output, that it will produce when run (or queried, or called upon). This lends monads to supplementing pure calculations with features like I/O, common environment or state, and to preprocessing of computations (simplification, optimization etc.).
Each monad, or computation type, provides means, subject to Monad Laws, to (a) create a description of computation action that will produce (a.k.a. "return") a given Haskell value, (b) somehow run a computation action description (possibly getting its output back into Haskell should the monad choose to allow it, if computations described by the monad are pure, or causing the prescribed side effects if it's not), and (c) combine (a.k.a. "bind") a computation action description with a reaction to it – a regular Haskell function of one argument (that will receive computation-produced value) returning another action description (using or dependent on that value, if need be) – thus creating a combined computation action description that will feed the original action's output through the reaction while automatically taking care of the particulars of the computational process itself. A monad might also define additional primitives to provide access to and/or enable manipulation of data it implicitly carries, specific to its nature.
Thus in Haskell, though it is a purely-functional language, side effects that will be performed by a computation can be dealt with and combined purely at the monad's composition time. Monads thus resemble programs in a particular DSL. While programs may describe impure effects and actions outside Haskell, they can still be combined and processed ("assembled") purely, inside Haskell, creating a pure Haskell value - a computation action description that describes an impure calculation. That is how Monads in Haskell separate between the pure and the impure.
The computation doesn't have to be impure and can be pure itself as well. Then monads serve to provide the benefits of separation of concerns, and automatic creation of a computational "pipeline". Because they are very useful in practice but rather mind-twisting for the beginners, numerous tutorials that deal exclusively with monads were created (see monad tutorials).
Contents |
1 Common monads
Most common applications of monads include:
- Representing failure using monadMaybe
- Nondeterminism using monad to represent carrying multiple valuesList
- State using monadState
- Read-only environment using monadReader
- I/O using monadIO
2 Monad class
Monads can be viewed as a standard programming interface to various data or control structures, which is captured by theclass Monad m where (>>=) :: m a -> (a -> m b) -> m b (>>) :: m a -> m b -> m b return :: a -> m a fail :: String -> m a
In addition to implementing the class functions, all instances of Monad should obey the following equations, or Monad Laws:
return a >>= k = k a m >>= return = m m >>= (\x -> k x >>= h) = (m >>= k) >>= h
See this intuitive explanation of why they should obey the Monad laws. It basically says that monad's reactions should be associative under Kleisli composition, defined as (f >=> g) x = f x >>= g
, with return
its left and right identity element.
Any Monad can be made a Functor by defining
fmap ab ma = ma >>= (return . ab)
However, the Functor class is not a superclass of the Monad class. See Functor hierarchy proposal.
3 Special notation
In order to improve the look of code that uses monads Haskell provides a special syntactic sugar calledthing1 >>= (\x -> func1 x >>= (\y -> thing2 >>= (\_ -> func2 y (\z -> return z))))
which can be written more clearly by breaking it into several lines and omitting parentheses:
thing1 >>= \x -> func1 x >>= \y -> thing2 >>= \_ -> func2 y >>= \z -> return z
do x <- thing1 y <- func1 x thing2 z <- func2 y return z
4 Commutative monads
Commutative monads are monads for which the order of actions makes no difference (they commute), that is when following code:
do a <- actA b <- actB m a b
is the same as:
do b <- actB a <- actA m a b
Examples of commutative include:
- monadReader
- monadMaybe
5 Monad tutorials
Monads are known for being deeply confusing to lots of people, so there are plenty of tutorials specifically related to monads. Each takes a different approach to Monads, and hopefully everyone will find something useful.
See the Monad tutorials timeline for a comprehensive list of monad tutorials.
6 Monad reference guides
An explanation of the basic Monad functions, with examples, can be found in the reference guide A tour of the Haskell Monad functions, by Henk-Jan van Tuyl.
7 Monad research
A collection of research papers about monads.
8 Monads in other languages
Implementations of monads in other languages.
- C
- C++, doc
- CML.event ?
- Clean State monad
- Clojure
- JavaScript
- Java
- Joy
- LINQ, more, C#, VB (inaccessible)
- Lisp
- Miranda
- OCaml:
- Perl
- Perl6 ?
- Prolog
- Python
- Python
- here
- Twisted's Deferred monad
- Ruby:
- Scala:
- Scheme:
- Tcl
- The Unix Shell
- More monads by Oleg
- CLL: a concurrent language based on a first-order intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
Unfinished:
- Slate
- Parsing, Maybe and Error in Tcl
And possibly there exist:
- Standard ML (via modules?)
Please add them if you know of other implementations.
Collection of links to monad implementations in various languages. on Lambda The Ultimate.
9 Interesting monads
A list of monads for various evaluation strategies and games:
- Identity monad
- Optional results
- Random values
- Read only state
- Writable state
- Unique supply
- ST - memory-only effects
- Global state
- Undoable state effects
- Function application
- Functions which may error
- Atomic memory transactions
- Continuations
- IO - unrestricted side effects
- Non-deterministic evaluation
- List monad: computations with multiple choices
- Concurrent threads
- Backtracking computations
- Region allocation effects
- LogicT: backtracking monad transformer with fair operations and pruning
- Pi calculus as a monad
- Halfs, uses a read-only and write-only monad for filesystem work.
- House's H monad for safe hardware access
- Commutable monads for parallel programming
- The Quantum computing monad
- Simple, Fair and Terminating Backtracking Monad
- Typed exceptions with call traces as a monad
- Breadth first list monad
- Continuation-based queues as monads
- Typed network protocol monad
- Non-Determinism Monad for Level-Wise Search
- Transactional state monad
- A constraint programming monad
- A probability distribution monad
There are many more interesting instance of the monad abstraction out there. Please add them as you come across each species.
10 Fun
- If you are tired of monads, you can easily get rid of them.