Difference between revisions of "Monad"
MichalPalka (talk  contribs) m (Added one more monad tutorial.) 
(small tweak) 

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{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=base}} 
{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=base}} 

−  '''Monads''' in Haskell are structures used to supplement pure computations with features like state, common environment or inputoutput. Even though Haskell is a purelyfunctional language, sideeffects can be conveniently simulated using monads. 

+  '''''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, updatable state, etc. 

−  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]]). 

+  Each monad, or computation type, provides means, subject to '''''Monad Laws''''', to 

−  == Common monads == 

+  * '''''(a)''''' ''create'' a description of a computation that will produce (a.k.a. "return") a given Haskell value, and 

+  * '''''(b)''''' ''combine'' (a.k.a. "bind") a computation description with a ''reaction'' to it, – a pure Haskell function that is set to receive a computationproduced value (when and if ''that'' happens) and return another computation description, using or dependent on that value if need be, – creating a description of a combined computation that will feed the original computation's output through the reaction while automatically taking care of the particulars of the computational process itself. 

+  
+  ''Reactions'' are thus computation description ''constructors''. A monad might also define additional primitives to provide access to and/or enable manipulation of data it implicitly carries, specific to its nature; cause some specific sideeffects; etc.. 

+  
+  Sometimes the specific monadic type also provides the ability to somehow '''''(c)''''' ''run'' a computation description, getting its result back into Haskell if computations described by the monad are pure, but this is expressly '''''not''''' a part of the Monad interface. Officially, <i>you can't get the <hask>a</hask> out of <hask>M a</hask></i> directly, only arrange for it to be "fed" into the next computation's constructor, the "reaction", indirectly. In case of an <hask>IO</hask> monad value, a computation it describes runs implicitly as a part of the chain of I/O computation descriptions composed together into the value <hask>main</hask> (of type <hask>IO ()</hask>) in a given Haskell program, by convention. <! Put simply, it runs when the compiled program runs (but then, everything does). > 

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

+  # Monad interactions: 

+  
+  (a) reaction $ value ==> computation_description 

+  
+  (b) reaction =<< computation_description ==> computation_description 

+  
+  (c) reaction $ computation_description ==> ***type_mismatch*** 

+  
+  (d) reaction <$> computation_description ==> computation_description_description 

+  
+  (e) join $ computation_description_description ==> computation_description 

+  </haskell> 

+  
+  (<i><hask>join</hask></i> is another function expressing the essence of monad; where <hask>m >>= k = k =<< m = join (k <$> m) = join (fmap k m)</hask>; it is prefered in mathematics, over the ''bind''; both express the same concept). 

+  
+  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 [[EDSL]] (''embedded'' domainspecific language, "embedded" because the values denoting these computations are legal Haskell values, not some extraneous annotations). 

+  
+  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 help keep the ''pure'' and the ''impure'' apart. 

+  
+  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]]). 

+  
+  == 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 using <hask>List</hask> monad to represent carrying multiple values 
* State using <hask>State</hask> monad 
* State using <hask>State</hask> monad 

* Readonly environment using <hask>Reader</hask> monad 
* Readonly environment using <hask>Reader</hask> monad 

−  * 
+  * I/O using <hask>IO</hask> monad 
== Monad class == 
== Monad class == 

−  All common monads are members of '''Monad''' class defined like this: 

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

<haskell> 
<haskell> 

class Monad m where 
class Monad m where 

−  (>>=) :: m a > (a > m b) > m b 
+  (>>=) :: m a > ( a > m b) > m b 
−  (>>) :: m a > m b > m b 
+  (>>) :: m a > m b > m b 
−  return :: a > m a 
+  return :: a > m a 
−  fail :: String > m a 
+  fail :: String > m a 
</haskell> 
</haskell> 

−  In addition to implementing the class functions, all instances of Monad should obey following equations: 
+  In addition to implementing the class functions, all instances of Monad should obey the following equations, or '''''Monad Laws''''': 
<haskell> 
<haskell> 

−  return a >>= k = k a 
+  return a >>= k = k a 
−  m >>= return = m 
+  m >>= return = m 
−  m >>= (\x > k x >>= h) = (m >>= k) >>= h 
+  m >>= (\x > k x >>= h) = (m >>= k) >>= h 
</haskell> 
</haskell> 

−  See [[Monad lawsthis intuitive explanation]] of why they should obey the Monad laws. 
+  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. 
−  Any Monad can be made a [[Functor]] by defining 

+  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. See [[Functor hierarchy proposal]]. 

+  
+  If the Monad definitions are preferred, Functor and Applicative 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 `return` as `pure`, if the two are the same. 

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

+  
<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 be also written using the <hask>do</hask>notation as follows: 

+  
+  This can also be written using the <hask>do</hask>notation as follows: 

+  
<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> 

−  Code written using the <hask>do</hask>notation is transformed to expressions that use <hask>Monad</hask> class functions by the compiler. 

−  When using the <hask>do</hask>notation and a monad like <hask>State</hask> or <hask>IO</hask> programs 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. 

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

+  
+  Code written using <hask>do</hask>notation is transformed by the compiler to ordinary expressions that use the functions from the <hask>Monad</hask> class (i.e. the two varieties of bind, <hask>>>=</hask> and <hask>>></hask>). 

+  
+  When using <hask>do</hask>notation and a monad like <hask>State</hask> or <hask>IO</hask> programs 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 <hask>do</hask>notation with regular notation. 
It is possible to intermix the <hask>do</hask>notation with regular notation. 

+  
+  More on <hask>do</hask>notation can be found in a section of [[Monads as computation#Do notationMonads as computation]] and in other [[Monad#Monad tutorialstutorials]]. 

+  
+  == Commutative monads == 

+  '''Commutative monads''' are monads for which the order of actions makes no difference (they '''commute'''), that is when following code: 

+  <haskell> 

+  do 

+  a < actA 

+  b < actB 

+  m a b 

+  </haskell> 

+  is the same as: 

+  <haskell> 

+  do 

+  b < actB 

+  a < actA 

+  m a b 

+  </haskell> 

+  
+  Examples of commutative include: 

+  * <hask>Reader</hask> monad 

+  * <hask>Maybe</hask> monad 

== Monad tutorials == 
== Monad tutorials == 

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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. 

−  * [http://www.engr.mun.ca/~theo/Misc/haskell_and_monads.htm Monads for the Working Haskell Programmer] 

+  See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials. 

−  * [[Monads as containers]], [[Monads as computation]] 

−  * [http://haskell.org/all_about_monads/html/index.html All About Monads] 

−  * [[Simple monad examples]] 

−  * [http://www.loria.fr/~kow/monads/index.html Of monads and space suits] 

−  * [http://sigfpe.blogspot.com/2006/08/youcouldhaveinventedmonadsand.html You could have invented monads] 

−  * [[Meet Bob The Monadic Lover]], or the slightly more serious [[The Monadic Way]] 

−  * Computational monads [http://programming.reddit.com/info/ox6s/comments/coxiv part 1] and [http://programming.reddit.com/info/ox6s/comments/coxoh part 2]. 

−  * [http://www.alpheccar.org/fr/posts/show/60 Three kind of monads] : sequencing, side effects or containers 

−  * [http://haskell.org/haskellwiki/Books_and_tutorials#Using_monads More tutorials on monads] 

−  * [http://haskell.org/haskellwiki/Blog_articles#Monads Even more tutorials on monads..] 

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

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Implementations of monads in other languages. 
Implementations of monads in other languages. 

−  * [http:// 
+  * [http://www.reddit.com/r/programming/comments/1761q/monads_in_c_pt_ii/ C] 
−  * [ 
+  * [https://github.com/clojure/algo.monads Clojure] 
* [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://cratylus.freewebspace.com/monadsinjavascript.htm JavaScript] 
* [http://cratylus.freewebspace.com/monadsinjavascript.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/ 
+  * [http://research.microsoft.com/enus/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ] 
−  * [http:// 
+  * [http://commonlisp.net/project/clmonadmacros/monadmacros.htm Lisp] 
* [http://lambdatheultimate.org/node/1136#comment12448 Miranda] 
* [http://lambdatheultimate.org/node/1136#comment12448 Miranda] 

* OCaml: 
* OCaml: 

** [http://www.cas.mcmaster.ca/~carette/pa_monad/ OCaml] 
** [http://www.cas.mcmaster.ca/~carette/pa_monad/ OCaml] 

** [https://mailman.rice.edu/pipermail/metaocamlusersl/2005March/000057.html more] 
** [https://mailman.rice.edu/pipermail/metaocamlusersl/2005March/000057.html more] 

−  ** [http://www.pps.jussieu.fr/~beffara/darcs/pivm/camlvm/monad.mli also] 

** [http://www.cas.mcmaster.ca/~carette/metamonads/ MetaOcaml] 
** [http://www.cas.mcmaster.ca/~carette/metamonads/ MetaOcaml] 

−  ** [http://enfranchisedmind.com 
+  ** [http://blog.enfranchisedmind.com/2007/08/amonadtutorialforocaml/ A Monad Tutorial for Ocaml] 
−  * [http:// 
+  * [http://www.reddit.com/r/programming/comments/p66e/are_monads_actually_used_in_anything_except Perl6 ?] 
−  * [http://programming.reddit.com/info/p66e/comments Perl6 ?] 

* [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Prolog] 
* [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Prolog] 

* Python 
* Python 

−  ** [http:// 
+  ** [http://code.activestate.com/recipes/439361/ Python] 
−  ** [http://www. 
+  ** Twisted's [http://www.reddit.com/r/programming/comments/p66e/are_monads_actually_used_in_anything_except/cp8eh Deferred monad] 
−  ** Twisted's [http://programming.reddit.com/info/p66e/comments/cp8eh Deferred monad] 

* Ruby: 
* Ruby: 

** [http://moonbase.rydia.net/mental/writings/programming/monadsinruby/00introduction.html Ruby] 
** [http://moonbase.rydia.net/mental/writings/programming/monadsinruby/00introduction.html Ruby] 

−  ** [http://metameta.blogspot.com/2006/12/monadsinrubypart1identity. 
+  ** [http://metameta.blogspot.com/2006/12/monadsinrubypart1identity.html and other implementation] 
−  * Scala: 

−  ** [http://scala.epfl.ch/examples/files/simpleInterpreter.html Scala] 

−  ** [http://scala.epfl.ch/examples/files/callccInterpreter.html A continuation monad] 

* Scheme: 
* Scheme: 

** [http://okmij.org/ftp/Scheme/monadinScheme.html Scheme] 
** [http://okmij.org/ftp/Scheme/monadinScheme.html Scheme] 

** [http://www.ccs.neu.edu/home/dherman/research/tutorials/monadsforschemers.txt also] 
** [http://www.ccs.neu.edu/home/dherman/research/tutorials/monadsforschemers.txt also] 

+  ** Monads & Do notation: [https://eltramo.be/blog/asyncmonad/ Part 1] [https://eltramo.be/blog/schememonads/ Part 2] 

+  * [http://www.javiersoto.me/post/106875422394 Swift] 

* [http://wiki.tcl.tk/13844 Tcl] 
* [http://wiki.tcl.tk/13844 Tcl] 

* [http://okmij.org/ftp/Computation/monadicshell.html The Unix Shell] 
* [http://okmij.org/ftp/Computation/monadicshell.html The Unix Shell] 

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Unfinished: 
Unfinished: 

−  * [http://slate.tunes.org/repos/main/src/unfinished/monad.slate Slate] 

* [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 

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Please add them if you know of other implementations. 
Please add them if you know of other implementations. 

−  [http://lambdatheultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambdatheultimate/ Lambda The Ultimate]. 
+  [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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A list of monads for various evaluation strategies and games: 
A list of monads for various evaluation strategies and games: 

−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadIdentity.html Identity monad]  the trivial monad. 
−  * [http://haskell.org/ghc/docs/latest/html/libraries/base/DataMaybe.html Optional results] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/DataMaybe.html Optional results from computations]  error checking without null. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/monadmersennerandom/latest/doc/html/ControlMonadMersenneRandom.html Random values]  run code in an environment with access to a stream of random numbers. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadReader.html Read only variables]  guarantee readonly access to values. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadWriterLazy.html Writable state]  i.e. log to a state buffer 
−  * [http://haskell.org/haskellwiki/New_monads/MonadSupply 
+  * [http://www.haskell.org/haskellwiki/New_monads/MonadSupply A supply of unique values]  useful for e.g. guids or unique variable names 
−  * [http://haskell.org/ghc/docs/latest/html/libraries/base/ControlMonadST.html ST] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/ControlMonadST.html ST  memoryonly, locallyencapsulated mutable variables]. Safely embed mutable state inside pure functions. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadStateLazy.html Global state]  a scoped, mutable state. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/Hedi/latest/doc/html/Undo.html Undoable state effects]  roll back state changes 
−  * [http://haskell.org/ghc/docs/latest/html/libraries/base/ControlMonadInstances.html Function application] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/ControlMonadInstances.html#t:Monad Function application]  chains of function application. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadError.html Functions which may error]  track location and causes of errors. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/stm/latest/doc/html/ControlMonadSTM.html Atomic memory transactions]  software transactional memory 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadCont.html Continuations]  computations which can be interrupted and resumed. 
−  * [http://haskell.org/ghc/docs/latest/html/libraries/base/SystemIO.html#t%3AIO IO] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/SystemIO.html#t%3AIO IO]  unrestricted side effects on the world 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/levelmonad/0.4.1/doc/html/ControlMonadLevels.html Search monad]  bfs and dfs search environments. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/streammonad/latest/doc/html/ControlMonadStream.html nondeterminism]  interleave computations with suspension. 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/stepwise/latest/doc/html/ControlMonadStepwise.html stepwise computation]  encode nondeterministic choices as stepwise deterministic ones 
−  * [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Backtracking] 
+  * [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Backtracking computations] 
−  * [http://www.cs.cornell.edu/people/fluet/research/rgnmonad/index.html Region allocation] 
+  * [http://www.cs.cornell.edu/people/fluet/research/rgnmonad/index.html Region allocation effects] 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/logict/0.5.0.2/doc/html/ControlMonadLogic.html LogicT]  backtracking monad transformer with fair operations and pruning 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/monadtask/latest/doc/html/ControlMonadTask.html concurrent events and threads]  refactor event and callback heavy programs into straightline code via coroutines 
−  * [http:// 
+  * [http://hackage.haskell.org/package/QIO QIO]  The Quantum computing monad 
−  * 
+  * [http://hackage.haskell.org/packages/archive/fullsessions/latest/doc/html/ControlConcurrentFullSession.html Pi calculus]  a monad for Picalculus style concurrent programming 
+  * [http://wwwfp.dcs.stand.ac.uk/~kh/papers/pasco94/subsubsectionstar3_3_2_3.html Commutable monads for parallel programming] 

+  * [http://hackage.haskell.org/package/streammonad Simple, Fair and Terminating Backtracking Monad] 

+  * [http://hackage.haskell.org/package/controlmonadexception Typed exceptions with call traces as a monad] 

+  * [http://hackage.haskell.org/package/controlmonadomega Breadth first list monad] 

+  * [http://hackage.haskell.org/package/controlmonadqueue Continuationbased queues as monads] 

+  * [http://hackage.haskell.org/package/fullsessions Typed network protocol monad] 

+  * [http://hackage.haskell.org/package/levelmonad NonDeterminism Monad for LevelWise Search] 

+  * [http://hackage.haskell.org/package/monadtx Transactional state monad] 

+  * [http://hackage.haskell.org/package/monadiccp A constraint programming monad] 

+  * [http://hackage.haskell.org/package/ProbabilityMonads A probability distribution monad] 

+  * [http://hackage.haskell.org/package/setmonad Sets]  Set computations 

+  * [http://hackage.haskell.org/package/httpmonad/ HTTP]  http connections as a monadic environment 

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

There are many more interesting instance of the monad abstraction out there. Please add them as you come across each species. 
There are many more interesting instance of the monad abstraction out there. Please add them as you come across each species. 

Line 177:  Line 228:  
==Fun== 
==Fun== 

−  * If you are tired of monads, you can easily [http:// 
+  * If you are tired of monads, you can easily [http://www.haskell.org.monadtransformer.parallelnetz.de/haskellwiki/Category:Monad 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/youcouldhaveinventedfreemonads.html Why free monads matter] (blog article) 

−  +  [[Category:Monad*]] 
Latest revision as of 11:05, 20 October 2020
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, updatable state, etc.
Each monad, or computation type, provides means, subject to Monad Laws, to
 (a) create a description of a computation that will produce (a.k.a. "return") a given Haskell value, and
 (b) combine (a.k.a. "bind") a computation description with a reaction to it, – a pure Haskell function that is set to receive a computationproduced value (when and if that happens) and return another computation description, using or dependent on that value if need be, – creating a description of a combined computation that will feed the original computation's output through the reaction while automatically taking care of the particulars of the computational process itself.
Reactions are thus computation description constructors. A monad might also define additional primitives to provide access to and/or enable manipulation of data it implicitly carries, specific to its nature; cause some specific sideeffects; etc..
Sometimes the specific monadic type also provides the ability to somehow (c) run a computation description, getting its result back into Haskell if computations described by the monad are pure, but this is expressly not a part of the Monad interface. Officially, you can't get the a
out of M a
directly, only arrange for it to be "fed" into the next computation's constructor, the "reaction", indirectly. In case of an IO
monad value, a computation it describes runs implicitly as a part of the chain of I/O computation descriptions composed together into the value main
(of type IO ()
) in a given Haskell program, by convention.
# Monad interactions:
(a) reaction $ value ==> computation_description
(b) reaction =<< computation_description ==> computation_description
(c) reaction $ computation_description ==> ***type_mismatch***
(d) reaction <$> computation_description ==> computation_description_description
(e) join $ computation_description_description ==> computation_description
(join
is another function expressing the essence of monad; where m >>= k = k =<< m = join (k <$> m) = join (fmap k m)
; it is prefered in mathematics, over the bind; both express the same concept).
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 EDSL (embedded domainspecific language, "embedded" because the values denoting these computations are legal Haskell values, not some extraneous annotations).
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 help keep the pure and the impure apart.
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 tutorials).
Contents
Common monads
Most common applications of monads 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
Monad class
Monads can be viewed as a standard programming interface to various data or control structures, which is captured by the Monad
class. All 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
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.
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. See 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 are the same.
do
notation
In order to improve the look of code that uses monads Haskell provides a special 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
This can also be written using the do
notation as follows:
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 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
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:

Reader
monad 
Maybe
monad
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.
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.
Unfinished:
 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.
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 instance 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.