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{{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 are structures used to supplement pure computations with features like state, common environment or I/O. Even though Haskell is a purely-functional language, side-effects 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 or state, and to ''preprocessing'' of computations (simplification, optimization etc.).
   
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]]).
+
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]]
  +
  +
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 through backtracking using <hask>List</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
 
* Read-only environment using <hask>Reader</hask> monad
 
* Read-only environment using <hask>Reader</hask> monad
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== 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>
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</haskell>
 
</haskell>
   
In addition to implementing the class functions, all instances of Monad should obey the following equations:
+
In addition to implementing the class functions, all instances of Monad should obey the following equations, or '''''Monad Laws''''':
   
 
<haskell>
 
<haskell>
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</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
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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:
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<haskell>
 
<haskell>
 
do
 
do
a <- f x
+
a <- actA
b <- g y
+
b <- actB
 
m a b
 
m a b
 
</haskell>
 
</haskell>
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<haskell>
 
<haskell>
 
do
 
do
b <- g y
+
b <- actB
a <- f x
+
a <- actA
 
m a b
 
m a b
 
</haskell>
 
</haskell>
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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.
   
See [[Tutorials#Using_monads|Monad tutorials]].
+
See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials.
   
 
== Monad reference guides ==
 
== Monad reference guides ==
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* [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://clean.cs.ru.nl/Download/Download_Libraries/Std_Env/StdFunc/stdfunc.html Clean] State monad
+
* [http://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf Clean] State monad
  +
* [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.tar.gz Java] (tar.gz)
+
* [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]
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** [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/archive/2007/08/06/307 A Monad Tutorial for Ocaml]
+
** [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 ?]
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* Python
 
* Python
 
** [http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/439361 Python]
 
** [http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/439361 Python]
** [http://www.python.org/pycon/papers/deferex/ here]
+
** [http://www.etsimo.uniovi.es/python/pycon/papers/deferex/ here]
 
** Twisted's [http://programming.reddit.com/info/p66e/comments/cp8eh Deferred monad]
 
** Twisted's [http://programming.reddit.com/info/p66e/comments/cp8eh Deferred monad]
 
* Ruby:
 
* Ruby:
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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/ghc/docs/latest/html/libraries/mtl/Control-Monad-Identity.html Identity monad]
+
* [http://hackage.haskell.org/packages/archive/mtl/1.1.0.2/doc/html/Control-Monad-Identity.html Identity monad]
 
* [http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Maybe.html Optional results]
 
* [http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Maybe.html Optional results]
 
* [http://haskell.org/haskellwiki/New_monads/MonadRandom Random values]
 
* [http://haskell.org/haskellwiki/New_monads/MonadRandom Random values]
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* [http://haskell.org/ghc/docs/latest/html/libraries/mtl/Control-Monad-Writer.html Writable state]
 
* [http://haskell.org/ghc/docs/latest/html/libraries/mtl/Control-Monad-Writer.html Writable state]
 
* [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]
+
* [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/mtl/Control-Monad-State.html State]
+
* [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]
+
* [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/ghc/docs/latest/html/libraries/mtl/Control-Monad-Error.html Error]
+
* [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/ghc/docs/latest/html/libraries/mtl/Control-Monad-Cont.html Continuations]
+
* [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]
+
* [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]
* [http://haskell.org/ghc/docs/latest/html/libraries/mtl/Control-Monad-List.html List monad]
+
* [http://haskell.org/ghc/docs/latest/html/libraries/mtl/Control-Monad-List.html List monad: computations with multiple choices]
 
* [http://www.math.chalmers.se/~koen/pubs/entry-jfp99-monad.html Concurrent threads]
 
* [http://www.math.chalmers.se/~koen/pubs/entry-jfp99-monad.html Concurrent threads]
* [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/rgn-monad/index.html Region allocation]
+
* [http://www.cs.cornell.edu/people/fluet/research/rgn-monad/index.html Region allocation effects]
 
* [http://okmij.org/ftp/Computation/monads.html#LogicT LogicT: backtracking monad transformer with fair operations and pruning]
 
* [http://okmij.org/ftp/Computation/monads.html#LogicT LogicT: backtracking monad transformer with fair operations and pruning]
 
* [http://tsukimi.agusa.i.is.nagoya-u.ac.jp/~sydney/PiMonad/ Pi calculus as a monad]
 
* [http://tsukimi.agusa.i.is.nagoya-u.ac.jp/~sydney/PiMonad/ Pi calculus as a monad]
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* House's H monad for safe hardware access
 
* House's H monad for safe hardware access
 
* [http://www-fp.dcs.st-and.ac.uk/~kh/papers/pasco94/subsubsectionstar3_3_2_3.html Commutable monads for parallel programming]
 
* [http://www-fp.dcs.st-and.ac.uk/~kh/papers/pasco94/subsubsectionstar3_3_2_3.html Commutable monads for parallel programming]
  +
* [http://hackage.haskell.org/package/QIO The Quantum computing monad]
  +
* [http://hackage.haskell.org/package/stream-monad Simple, Fair and Terminating Backtracking Monad]
  +
* [http://hackage.haskell.org/package/control-monad-exception Typed exceptions with call traces as a monad]
  +
* [http://hackage.haskell.org/package/control-monad-omega Breadth first list monad]
  +
* [http://hackage.haskell.org/package/control-monad-queue Continuation-based queues as monads]
  +
* [http://hackage.haskell.org/package/full-sessions Typed network protocol monad]
  +
* [http://hackage.haskell.org/package/level-monad Non-Determinism Monad for Level-Wise Search]
  +
* [http://hackage.haskell.org/package/monad-tx Transactional state monad]
  +
* [http://hackage.haskell.org/package/monadiccp A constraint programming monad]
  +
* [http://hackage.haskell.org/package/ProbabilityMonads 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.
 
There are many more interesting instance of the monad abstraction out there. Please add them as you come across each species.
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* 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].
   
[[Category:Standard classes]] [[Category:Monad]]
+
==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|*]]

Revision as of 22:09, 3 September 2012

Monad class (base)
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.

Monads inter-dependencies 2.png

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
    Maybe
    monad
  • Nondeterminism using
    List
    monad to represent carrying multiple values
  • State using
    State
    monad
  • Read-only environment using
    Reader
    monad
  • I/O using
    IO
    monad

2 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.

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 called
do
-notation. For example, 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 be also written using the
do
-notation as follows:
do
  x <- thing1
  y <- func1 x
  thing2
  z <- func2 y
  return z
Code written using the
do
-notation is transformed by the compiler to ordinary expressions that use
Monad
class functions. When using the
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 the
do
-notation can be found in a section of Monads as computation and in other tutorials.

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:

  • Reader
    monad
  • Maybe
    monad

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.

Unfinished:

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:


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

10 Fun

11 See also