Difference between revisions of "Monad"
(→Monad class: improved indentation) 
(fix link to erik meijer linq webpage) 

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+  ''Hint: if you're just looking for an introduction to monads, see [[Merely monadic]] or one of the other [[Monad tutorials timelinemonad tutorials]].'' 

+   

+  
{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=base}} 
{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=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, 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, 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 computationproduced 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. 

−  
−  <haskell style="backgroundcolor:#f8f1ab;borderradius:15px;border:2px solid #000000;padding:15px"> 

−  # Monad Laws in pseudo code 

−  (a) reaction $ value ==> action_description 

−  
−  (c) action_description `bind` reaction ==> action_description 

−  
−  (b) run action_description ==> value 

−  </haskell> 

−  
−  Thus in Haskell, though it is a purelyfunctional 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 mindtwisting for the beginners, numerous tutorials that deal exclusively with monads were created (see [[Monad#Monad tutorialsmonad tutorials]]). 

−  
⚫  
−  Most common applications of monads include: 

⚫  
⚫  
⚫  
⚫  
⚫  
−  
⚫  
−  Monads can be viewed as a standard programming interface to various data or control structures, which is captured by 
+  Monads can be viewed as a standard programming interface to various data or control structures, which is captured by Haskell's <code>Monad</code> class. All the common monads are members of it: 
<haskell> 
<haskell> 

Line 35:  Line 13:  
(>>) :: m a > m b > m b 
(>>) :: m a > m b > m b 

return :: a > m a 
return :: a > m a 

−  fail :: String > m a 

</haskell> 
</haskell> 

−  In addition to implementing the class functions, all instances of Monad should 
+  In addition to implementing the class functions, all instances of <code>Monad</code> should satisfy the following equations, or ''monad laws'': 
<haskell> 
<haskell> 

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</haskell> 
</haskell> 

−  See [[Monad lawsthis 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. 

+  For more information, including an intuitive explanation of why the monad laws should be satisfied, see [[Monad laws]]. 

−  Any Monad can be made a [[Functor]] by defining 

+  As of GHC 7.10, the <code>Applicative</code> typeclass is a superclass of <code>Monad</code>, and the <code>Functor</code> typeclass is a superclass of <code>Applicative</code>. This means that all monads are applicatives, all applicatives are functors, and therefore all monads are also functors. For more information, see the [[Functor hierarchy proposal]]. 

+  
+  If the <code>Monad</code> definitions are preferred, <code>Functor</code> and <code>Applicative</code> instances can be defined from them with: 

<haskell> 
<haskell> 

−  fmap 
+  fmap fab ma = do { a < ma ; return (fab a) } 
+   ma >>= (return . fab) 

+  pure a = do { return a } 

+   return a 

+  mfab <*> ma = do { fab < mfab ; a < ma ; return (fab a) } 

+   mfab >>= (\ fab > ma >>= (return . fab)) 

+   mfab `ap` ma 

</haskell> 
</haskell> 

−  However, the Functor class is not a superclass of the Monad class. See [[Functor hierarchy proposal]]. 

+  although the recommended order is to define <code>return</code> as <code>pure</code> if the two would otherwise end up being the same. 

+  
⚫  
+  These include: 

⚫  
⚫  
⚫  
⚫  
⚫  
−  == 
+  == <code>do</code>notation == 
−  In order to improve the look of code that uses monads Haskell provides a special [[syntactic sugar]] called < 
+  In order to improve the look of code that uses monads, Haskell provides a special form of [[syntactic sugar]] called <code>do</code>notation. For example, the following expression: 
<haskell> 
<haskell> 

Line 68:  Line 55:  
<haskell> 
<haskell> 

−  thing1 >>= \x > 
+  thing1 >>= \x > 
func1 x >>= \y > 
func1 x >>= \y > 

−  thing2 >>= \_ > 
+  thing2 >>= \_ > 
func2 y >>= \z > 
func2 y >>= \z > 

return z 
return z 

</haskell> 
</haskell> 

−  +  can also be written using <code>do</code>notation: 

<haskell> 
<haskell> 

−  do 
+  do { 
−  x < thing1 
+  x < thing1 ; 
−  y < func1 x 
+  y < func1 x ; 
−  thing2 
+  thing2 ; 
−  z < func2 y 
+  z < func2 y ; 
return z 
return z 

+  } 

</haskell> 
</haskell> 

⚫  
+  (the curly braces and the semicolons are optional when the indentation rules are observed). 

⚫  When using < 

⚫  
⚫  
⚫  When using <code>do</code>notation and a monad like <code>State</code> or <code>IO</code>, programs in Haskell look very much like programs written in an imperative language as each line contains a statement that can change the simulated global state of the program and optionally binds a (local) variable that can be used by the statements later in the code block. 

⚫  
⚫  
+  
⚫  
== Commutative monads == 
== Commutative monads == 

−  +  For monads which are ''commutative'' the order of actions makes no difference (i.e. they ''commute''), so the following code: 

<haskell> 
<haskell> 

do 
do 

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</haskell> 
</haskell> 

−  Examples of commutative include: 
+  Examples of commutative monads include: 
−  * < 
+  * <code>Reader</code> monad 
−  * < 
+  * <code>Maybe</code> monad 
== Monad tutorials == 
== Monad tutorials == 

−  Monads are known for being 
+  Monads are known for being quite confusing to many 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. 
See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials. 

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== Monad reference guides == 
== Monad reference guides == 

−  An explanation of the basic Monad functions, with examples, can be found in the reference guide [http://members.chello.nl/hjgtuyl/tourdemonad.html A tour of the Haskell Monad functions] 
+  An explanation of the basic <code>Monad</code> functions, with examples, can be found in the reference guide [http://members.chello.nl/hjgtuyl/tourdemonad.html A tour of the Haskell Monad functions] by HenkJan van Tuyl. 
== Monad research == 
== Monad research == 

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* [http://www.ccs.neu.edu/home/dherman/browse/code/monads/JavaMonads/ Java] 
* [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/enus/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ] 
+  * [https://web.archive.org/web/20130522092554/http://research.microsoft.com/enus/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ] 
* [http://commonlisp.net/project/clmonadmacros/monadmacros.htm Lisp] 
* [http://commonlisp.net/project/clmonadmacros/monadmacros.htm Lisp] 

* [http://lambdatheultimate.org/node/1136#comment12448 Miranda] 
* [http://lambdatheultimate.org/node/1136#comment12448 Miranda] 

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* [http://okmij.org/ftp/Computation/monads.html More monads by Oleg] 
* [http://okmij.org/ftp/Computation/monads.html More monads by Oleg] 

* [http://lambdatheultimate.org/node/2322 CLL]: a concurrent language based on a firstorder intuitionistic linear logic where all right synchronous connectives are restricted to a monad. 
* [http://lambdatheultimate.org/node/2322 CLL]: a concurrent language based on a firstorder intuitionistic linear logic where all right synchronous connectives are restricted to a monad. 

+  * [http://lambdatheultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambdatheultimate.org/ Lambda The Ultimate]. 

Unfinished: 
Unfinished: 

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* [http://wiki.tcl.tk/14295 Parsing], [http://wiki.tcl.tk/13844 Maybe and Error] in Tcl 
* [http://wiki.tcl.tk/14295 Parsing], [http://wiki.tcl.tk/13844 Maybe and Error] in Tcl 

−  And possibly there 
+  And possibly there exists: 
* Standard ML (via modules?) 
* Standard ML (via modules?) 

−  +  ''(If you know of other implementations, please add them here.)'' 

−  
−  [http://lambdatheultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambdatheultimate.org/ Lambda The Ultimate]. 

==Interesting monads== 
==Interesting monads== 

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* [http://hackage.haskell.org/package/monadmemo Memoization]  add memoization to code 
* [http://hackage.haskell.org/package/monadmemo Memoization]  add memoization to code 

−  There are many more interesting 
+  There are many more interesting instances of the monad abstraction out there. Please add them as you come across each species. 
==Fun== 
==Fun== 

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[[Category:Monad*]] 
[[Category:Monad*]] 

+  [[Category:Nondeterminism]] 
Latest revision as of 16:30, 1 August 2021
Hint: if you're just looking for an introduction to monads, see Merely monadic or one of the other monad tutorials.
import Control.Monad 
Contents
The Monad
class
Monads can be viewed as a standard programming interface to various data or control structures, which is captured by Haskell's Monad
class. All the common monads are members of it:
class Monad m where
(>>=) :: m a > ( a > m b) > m b
(>>) :: m a > m b > m b
return :: a > m a
In addition to implementing the class functions, all instances of Monad
should satisfy the following equations, or monad laws:
return a >>= k = k a
m >>= return = m
m >>= (\x > k x >>= h) = (m >>= k) >>= h
For more information, including an intuitive explanation of why the monad laws should be satisfied, see Monad laws.
As of GHC 7.10, the Applicative
typeclass is a superclass of Monad
, and the Functor
typeclass is a superclass of Applicative
. This means that all monads are applicatives, all applicatives are functors, and therefore all monads are also functors. For more information, see the Functor hierarchy proposal.
If the Monad
definitions are preferred, Functor
and Applicative
instances can be defined from them with:
fmap fab ma = do { a < ma ; return (fab a) }
 ma >>= (return . fab)
pure a = do { return a }
 return a
mfab <*> ma = do { fab < mfab ; a < ma ; return (fab a) }
 mfab >>= (\ fab > ma >>= (return . fab))
 mfab `ap` ma
although the recommended order is to define return
as pure
if the two would otherwise end up being the same.
Common monads
These include:
 Representing failure using
Maybe
monad  Nondeterminism using
List
monad to represent carrying multiple values  State using
State
monad  Readonly environment using
Reader
monad  I/O using
IO
monad
do
notation
In order to improve the look of code that uses monads, Haskell provides a special form of syntactic sugar called do
notation. For example, the following expression:
thing1 >>= (\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
can also be written using do
notation:
do {
x < thing1 ;
y < func1 x ;
thing2 ;
z < func2 y ;
return z
}
(the curly braces and the semicolons are optional when the indentation rules are observed).
Code written using do
notation is transformed by the compiler to ordinary expressions that use the functions from the Monad
class (i.e. the two varieties of bind: (>>=)
and (>>)
).
When using do
notation and a monad like State
or IO
, programs in Haskell look very much like programs written in an imperative language as each line contains a statement that can change the simulated global state of the program and optionally binds a (local) variable that can be used by the statements later in the code block.
It is possible to intermix the do
notation with regular notation.
More on do
notation can be found in a section of Monads as computation and in other tutorials.
Commutative monads
For monads which are commutative the order of actions makes no difference (i.e. they commute), so the 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 monads include:

Reader
monad 
Maybe
monad
Monad tutorials
Monads are known for being quite confusing to many 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.
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 HenkJan van Tuyl.
Monad research
A collection of research papers about monads.
Monads in other languages
Implementations of monads in other languages.
 C
 Clojure
 CML.event ?
 Clean State monad
 JavaScript
 Java
 Joy
 LINQ
 Lisp
 Miranda
 OCaml:
 Perl6 ?
 Prolog
 Python
 Python
 Twisted's Deferred monad
 Ruby:
 Scheme:
 Swift
 Tcl
 The Unix Shell
 More monads by Oleg
 CLL: a concurrent language based on a firstorder intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
 Collection of links to monad implementations in various languages. on Lambda The Ultimate.
Unfinished:
 Parsing, Maybe and Error in Tcl
And possibly there exists:
 Standard ML (via modules?)
(If you know of other implementations, please add them here.)
Interesting monads
A list of monads for various evaluation strategies and games:
 Identity monad  the trivial monad.
 Optional results from computations  error checking without null.
 Random values  run code in an environment with access to a stream of random numbers.
 Read only variables  guarantee readonly access to values.
 Writable state  i.e. log to a state buffer
 A supply of unique values  useful for e.g. guids or unique variable names
 ST  memoryonly, locallyencapsulated mutable variables. Safely embed mutable state inside pure functions.
 Global state  a scoped, mutable state.
 Undoable state effects  roll back state changes
 Function application  chains of function application.
 Functions which may error  track location and causes of errors.
 Atomic memory transactions  software transactional memory
 Continuations  computations which can be interrupted and resumed.
 IO  unrestricted side effects on the world
 Search monad  bfs and dfs search environments.
 nondeterminism  interleave computations with suspension.
 stepwise computation  encode nondeterministic choices as stepwise deterministic ones
 Backtracking computations
 Region allocation effects
 LogicT  backtracking monad transformer with fair operations and pruning
 concurrent events and threads  refactor event and callback heavy programs into straightline code via coroutines
 QIO  The Quantum computing monad
 Pi calculus  a monad for Picalculus style concurrent programming
 Commutable monads for parallel programming
 Simple, Fair and Terminating Backtracking Monad
 Typed exceptions with call traces as a monad
 Breadth first list monad
 Continuationbased queues as monads
 Typed network protocol monad
 NonDeterminism Monad for LevelWise Search
 Transactional state monad
 A constraint programming monad
 A probability distribution monad
 Sets  Set computations
 HTTP  http connections as a monadic environment
 Memoization  add memoization to code
There are many more interesting instances of the monad abstraction out there. Please add them as you come across each species.
Fun
 If you are tired of monads, you can easily get rid of them.