Difference between revisions of "Monad"

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(Added proper introduction, moved the content about Monad class to its own section, added a section about the do-notation.)
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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 timeline|monad tutorials]].''
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----
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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 input-output. Even though Haskell is a purely-functional language, side-effects can be conveniently simulated using monads.
 
  +
== The <code>Monad</code> class ==
 
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 ==
 
 
Most common applications of monads include:
 
* Representing failure using <hask>Maybe</hask> monad
 
* Non-determinism using <hask>List</hask> monad
 
* State using <hask>State</hask> monad
 
* Read-only environment using <hask>Reader</hask> monad
 
* Input-output using <hask>IO</hask> monad
 
   
== Monad class ==
 
  +
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:
All common monads are members of '''Monad''' class defined like this:
 
   
 
<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
 
 
</haskell>
 
</haskell>
   
In addition to implementing the class functions, all instances of Monad should obey following equations:
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In addition to implementing the class functions, all instances of <code>Monad</code> should satisfy 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 laws|this intuitive explanation]] of why they should obey the Monad laws.
+
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 ab ma = ma >>= (return . ab)
+
fmap fab ma = do { a <- ma ; return (fab a) }
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-- 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.
   
== Simplified notation ==
+
== Common monads ==
  +
These include:
  +
* Representing failure using <code>Maybe</code> monad
  +
* Nondeterminism using <code>List</code> monad to represent carrying multiple values
  +
* State using <code>State</code> monad
  +
* Read-only environment using <code>Reader</code> monad
  +
* I/O using <code>IO</code> monad
  +
  +
== <code>do</code>-notation ==
  +
  +
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:
   
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:
  +
 
<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:
 
  +
  +
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>
   
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).
   
It is possible to intermix the <hask>do</hask>-notation with regular notation.
 
  +
Code written using <code>do</code>-notation is transformed by the compiler to ordinary expressions that use the functions from the <code>Monad</code> class (i.e. the two varieties of bind: <code>(>>=)</code> and <code>(>>)</code>).
  +
  +
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.
  +
  +
It is possible to intermix the <code>do</code>-notation with regular notation.
  +
  +
More on <code>do</code>-notation can be found in a section of [[Monads as computation#Do notation|Monads as computation]] and in other [[Monad#Monad tutorials|tutorials]].
  +
  +
== Commutative monads ==
  +
For monads which are ''commutative'' the order of actions makes no difference (i.e. they ''commute''), so the 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 monads include:
  +
* <code>Reader</code> monad
  +
* <code>Maybe</code> monad
   
 
== Monad tutorials ==
 
== 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.
+
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.
   
* [[Monads as containers]], [[Monads as computation]]
 
  +
See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials.
* [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/you-could-have-invented-monads-and.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 ==
   
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], by Henk-Jan van Tuyl.
+
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 Henk-Jan van Tuyl.
   
 
== Monad research ==
 
== Monad research ==
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Implementations of monads in other languages.
 
Implementations of monads in other languages.
   
* [http://programming.reddit.com/goto?id=1761q C]
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* [http://www.reddit.com/r/programming/comments/1761q/monads_in_c_pt_ii/ C]
* [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]
+
* [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://clean.cs.ru.nl/Download/Download_Libraries/Std_Env/StdFunc/stdfunc.html Clean] State monad
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* [http://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf Clean] State monad
 
* [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/en-us/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ]
* [http://sleepingsquirrel.org/monads/monads.lisp Lisp]
+
* [http://common-lisp.net/project/cl-monad-macros/monad-macros.htm Lisp]
 
* [http://lambda-the-ultimate.org/node/1136#comment-12448 Miranda]
 
* [http://lambda-the-ultimate.org/node/1136#comment-12448 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/metaocaml-users-l/2005-March/000057.html more]
 
** [https://mailman.rice.edu/pipermail/metaocaml-users-l/2005-March/000057.html more]
** [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://blog.enfranchisedmind.com/2007/08/a-monad-tutorial-for-ocaml/ A Monad Tutorial for Ocaml]
* [http://sleepingsquirrel.org/monads/monads.html Perl]
+
* [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://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/439361 Python]
+
** [http://code.activestate.com/recipes/439361/ Python]
** [http://www.python.org/pycon/papers/deferex/ here]
+
** 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/monads-in-ruby/00introduction.html Ruby]
 
** [http://moonbase.rydia.net/mental/writings/programming/monads-in-ruby/00introduction.html Ruby]
** [http://meta-meta.blogspot.com/2006/12/monads-in-ruby-part-1-identity.htmland other implementation]
+
** [http://meta-meta.blogspot.com/2006/12/monads-in-ruby-part-1-identity.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/monad-in-Scheme.html Scheme]
 
** [http://okmij.org/ftp/Scheme/monad-in-Scheme.html Scheme]
 
** [http://www.ccs.neu.edu/home/dherman/research/tutorials/monads-for-schemers.txt also]
 
** [http://www.ccs.neu.edu/home/dherman/research/tutorials/monads-for-schemers.txt also]
  +
** Monads & Do notation: [https://el-tramo.be/blog/async-monad/ Part 1] [https://el-tramo.be/blog/scheme-monads/ 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/monadic-shell.html The Unix Shell]
 
* [http://okmij.org/ftp/Computation/monadic-shell.html The Unix Shell]
 
* [http://okmij.org/ftp/Computation/monads.html More monads by Oleg]
 
* [http://okmij.org/ftp/Computation/monads.html More monads by Oleg]
 
* [http://lambda-the-ultimate.org/node/2322 CLL]: a concurrent language based on a first-order intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
 
* [http://lambda-the-ultimate.org/node/2322 CLL]: a concurrent language based on a first-order intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
  +
* [http://lambda-the-ultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambda-the-ultimate.org/ Lambda The Ultimate].
   
 
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
   
And possibly there exist:
+
And possibly there exists:
   
 
* Standard ML (via modules?)
 
* Standard ML (via modules?)
   
Please add them if you know of other implementations.
+
''(If you know of other implementations, please add them here.)''
 
[http://lambda-the-ultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambda-the-ultimate/ 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/ghc/docs/latest/html/libraries/mtl/Control-Monad-Identity.html Identity monad]
+
* [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Identity.html Identity monad] - the trivial monad.
* [http://haskell.org/ghc/docs/latest/html/libraries/base/Data-Maybe.html Optional results]
+
* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Data-Maybe.html Optional results from computations] - error checking without null.
* [http://haskell.org/haskellwiki/New_monads/MonadRandom Random values]
+
* [http://hackage.haskell.org/packages/archive/monad-mersenne-random/latest/doc/html/Control-Monad-Mersenne-Random.html Random values] - run code in an environment with access to a stream of random numbers.
* [http://haskell.org/ghc/docs/latest/html/libraries/mtl/Control-Monad-Reader.html Read only state]
+
* [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Reader.html Read only variables] - guarantee read-only access to values.
* [http://haskell.org/ghc/docs/latest/html/libraries/mtl/Control-Monad-Writer.html Writable state]
+
* [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Writer-Lazy.html Writable state] - i.e. log to a state buffer
* [http://haskell.org/haskellwiki/New_monads/MonadSupply Unique supply]
+
* [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/Control-Monad-ST.html ST]
+
* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-ST.html ST - memory-only, locally-encapsulated mutable variables]. Safely embed mutable state inside pure functions.
* [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-Lazy.html Global state] - a scoped, mutable state.
* [http://haskell.org/haskellwiki/New_monads/MonadUndo Undoable state]
+
* [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/Control-Monad-Instances.html Function application]
+
* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-Instances.html#t:Monad Function application] - chains of 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] - track location and causes of errors.
* [http://haskell.org/ghc/docs/latest/html/libraries/stm/Control-Monad-STM.html Atomic memory transactions]
+
* [http://hackage.haskell.org/packages/archive/stm/latest/doc/html/Control-Monad-STM.html Atomic memory transactions] - software transactional memory
* [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] - computations which can be interrupted and resumed.
* [http://haskell.org/ghc/docs/latest/html/libraries/base/System-IO.html#t%3AIO IO]
+
* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/System-IO.html#t%3AIO IO] - unrestricted side effects on the world
* [http://www.haskell.org/haskellwiki/Sudoku Non-deterministic evaluation]
+
* [http://hackage.haskell.org/packages/archive/level-monad/0.4.1/doc/html/Control-Monad-Levels.html Search monad] - bfs and dfs search environments.
* [http://haskell.org/ghc/docs/latest/html/libraries/mtl/Control-Monad-List.html List monad]
+
* [http://hackage.haskell.org/packages/archive/stream-monad/latest/doc/html/Control-Monad-Stream.html non-determinism] - interleave computations with suspension.
* [http://www.math.chalmers.se/~koen/pubs/entry-jfp99-monad.html Concurrent threads]
+
* [http://hackage.haskell.org/packages/archive/stepwise/latest/doc/html/Control-Monad-Stepwise.html stepwise computation] - encode non-deterministic 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/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://hackage.haskell.org/packages/archive/logict/0.5.0.2/doc/html/Control-Monad-Logic.html 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://hackage.haskell.org/packages/archive/monad-task/latest/doc/html/Control-Monad-Task.html concurrent events and threads] - refactor event and callback heavy programs into straight-line code via co-routines
* [http://www.haskell.org/halfs/ Halfs], uses a read-only and write-only monad for filesystem work.
+
* [http://hackage.haskell.org/package/QIO QIO] - The Quantum computing monad
* House's H monad for safe hardware access
+
* [http://hackage.haskell.org/packages/archive/full-sessions/latest/doc/html/Control-Concurrent-FullSession.html Pi calculus] - a monad for Pi-calculus style concurrent 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/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]
  +
* [http://hackage.haskell.org/package/set-monad Sets] - Set computations
  +
* [http://hackage.haskell.org/package/http-monad/ HTTP] - http connections as a monadic environment
  +
* [http://hackage.haskell.org/package/monad-memo 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 instances of the monad abstraction out there. Please add them as you come across each species.
   
 
==Fun==
 
==Fun==
   
* 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://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/you-could-have-invented-free-monads.html Why free monads matter] (blog article)
   
[[Category:Standard classes]] [[Category:Monad]]
+
[[Category:Monad|*]]
  +
[[Category:Nondeterminism]]

Latest revision as of 23:15, 14 June 2021

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


Monad class (base)
import Control.Monad

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
  • Read-only 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 Henk-Jan van Tuyl.

Monad research

A collection of research papers about monads.

Monads in other languages

Implementations of monads in other languages.

Unfinished:

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:

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

Fun

See also