Difference between revisions of "Monad"

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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]].''
  +
----
  +
 
{{Standard class|Monad|module=Control.Monad|module-doc=Control-Monad|package=base}}
 
{{Standard class|Monad|module=Control.Monad|module-doc=Control-Monad|package=base}}
The '''Monad''' class is defined like this:
 
  +
  +
== The <code>Monad</code> 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:
   
 
<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>
   
All instances of Monad should obey:
 
  +
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>
   
Any Monad can be made a [[Functor]] by defining
 
  +
For more information, including an intuitive explanation of why the monad laws should be satisfied, see [[Monad laws]].
  +
  +
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) }
  +
-- 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.
  +
  +
== 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:
  +
  +
<haskell>
  +
thing1 >>= (\x -> func1 x >>= (\y -> thing2
  +
>>= (\_ -> func2 y >>= (\z -> return z))))
  +
</haskell>
  +
  +
which can be written more clearly by breaking it into several lines and omitting parentheses:
  +
  +
<haskell>
  +
thing1 >>= \x ->
  +
func1 x >>= \y ->
  +
thing2 >>= \_ ->
  +
func2 y >>= \z ->
  +
return z
  +
</haskell>
  +
  +
can also be written using <code>do</code>-notation:
  +
  +
<haskell>
  +
do {
  +
x <- thing1 ;
  +
y <- func1 x ;
  +
thing2 ;
  +
z <- func2 y ;
  +
return z
  +
}
  +
</haskell>
  +
  +
(the curly braces and the semicolons are optional when the indentation rules are observed).
  +
  +
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 ==
  +
 
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 <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 ==
  +
  +
A collection of [[Research_papers/Monads_and_arrows|research papers]] about monads.
  +
  +
== Monads in other languages ==
  +
  +
Implementations of monads in other languages.
  +
  +
* [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://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf Clean] State monad
  +
* [http://cratylus.freewebspace.com/monads-in-javascript.htm JavaScript]
  +
* [http://www.ccs.neu.edu/home/dherman/browse/code/monads/JavaMonads/ Java]
  +
* [http://permalink.gmane.org/gmane.comp.lang.concatenative/1506 Joy]
  +
* [https://web.archive.org/web/20130522092554/http://research.microsoft.com/en-us/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ]
  +
* [http://common-lisp.net/project/cl-monad-macros/monad-macros.htm Lisp]
  +
* [http://lambda-the-ultimate.org/node/1136#comment-12448 Miranda]
  +
* OCaml:
  +
** [http://www.cas.mcmaster.ca/~carette/pa_monad/ OCaml]
  +
** [https://mailman.rice.edu/pipermail/metaocaml-users-l/2005-March/000057.html more]
  +
** [http://www.cas.mcmaster.ca/~carette/metamonads/ MetaOcaml]
  +
** [http://blog.enfranchisedmind.com/2007/08/a-monad-tutorial-for-ocaml/ A Monad Tutorial for Ocaml]
  +
* [http://www.reddit.com/r/programming/comments/p66e/are_monads_actually_used_in_anything_except Perl6 ?]
  +
* [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Prolog]
  +
* Python
  +
** [http://code.activestate.com/recipes/439361/ Python]
  +
** Twisted's [http://www.reddit.com/r/programming/comments/p66e/are_monads_actually_used_in_anything_except/cp8eh Deferred monad]
  +
* 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.html and other implementation]
  +
* 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]
  +
** 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://okmij.org/ftp/Computation/monadic-shell.html The Unix Shell]
 
* [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/1136 Collection of links to monad implementations in various languages.] on [http://lambda-the-ultimate.org/ Lambda The Ultimate].
  +
  +
Unfinished:
  +
  +
* [http://wiki.tcl.tk/14295 Parsing], [http://wiki.tcl.tk/13844 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:
   
== Monad Tutorials ==
 
  +
* [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Identity.html Identity monad] - the trivial monad.
  +
* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Data-Maybe.html Optional results from computations] - error checking without null.
  +
* [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://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Reader.html Read only variables] - guarantee read-only access to values.
  +
* [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://www.haskell.org/haskellwiki/New_monads/MonadSupply A supply of unique values] - useful for e.g. guids or unique variable names
  +
* [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://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-State-Lazy.html Global state] - a scoped, mutable state.
  +
* [http://hackage.haskell.org/packages/archive/Hedi/latest/doc/html/Undo.html Undoable state effects] - roll back state changes
  +
* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Control-Monad-Instances.html#t:Monad Function application] - chains of function application.
  +
* [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://hackage.haskell.org/packages/archive/stm/latest/doc/html/Control-Monad-STM.html Atomic memory transactions] - software transactional memory
  +
* [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Cont.html Continuations] - computations which can be interrupted and resumed.
  +
* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/System-IO.html#t%3AIO IO] - unrestricted side effects on the world
  +
* [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://hackage.haskell.org/packages/archive/stream-monad/latest/doc/html/Control-Monad-Stream.html non-determinism] - interleave computations with suspension.
  +
* [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 computations]
  +
* [http://www.cs.cornell.edu/people/fluet/research/rgn-monad/index.html Region allocation effects]
  +
* [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://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://hackage.haskell.org/package/QIO QIO] - The Quantum computing monad
  +
* [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
   
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.
 
  +
There are many more interesting instances of the monad abstraction out there. Please add them as you come across each species.
   
* [[Monads as Containers]]
 
  +
==Fun==
* [http://www.nomaware.com/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]]
 
   
== Monad Reference Guides ==
 
  +
* If you are tired of monads, you can easily [http://www.haskell.org.monadtransformer.parallelnetz.de/haskellwiki/Category:Monad get rid of them].
   
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.
 
  +
==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: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.


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