Difference between revisions of "Monad"

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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 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. This lends monads to supplementing ''pure'' calculations with features like I/O, common environment or state, and to ''preprocessing'' of computations (simplification, optimization etc.).
+
'''''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''''', of (a) creating a description of computation to produce a ''given value'' (or such that will ''fail'' to produce anything at all), (b) ''running'' a computation description (CD) and returning its output to Haskell, and (c) combining a CD with a ''"reaction"'' to it, i.e. a Haskell function consuming of its output and returning another CD (using or dependent on that output, if need be), to create a ''combined'' CD. It might also define additional primitives to provide access and/or enable manipulation of data it implicitly carries, specific to its nature.
 
  +
Each monad, or computation type, provides means, subject to '''''Monad Laws''''', to
   
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 CD that describes an impure calculation. That is how Monads in Haskell '''''separate''''' between the ''pure'' and the ''impure''.
 
  +
* '''''(a)''''' ''create'' a description of a computation that will produce (a.k.a. "return") a given Haskell value, and
   
The computation doesn't have to be impure and can be pure itself as well. Then Monads serve to separate the pure from the pure in one big holiday celebration after the other. We still get the benefits of separation of concerns, and automatic creation of a computational "pipeline" carrying out our chained Haskell calculations one after another with computation's state threaded through behind the scenes.
 
  +
* '''''(b)''''' ''combine'' (a.k.a. "bind") a computation description with a ''reaction'' to it, – a pure Haskell function that is set to receive a computation-produced 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.
   
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]]).
 
  +
''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 side-effects; etc..
  +
  +
Sometimes the specific monadic type also provides the ability to somehow ''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="background-color:#f8f1ab;border-radius: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 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 [[EDSL]] (''embedded'' domain-specific 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 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
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* 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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<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>
   
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<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. 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.
 
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
 
  +
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 ab ma = ma >>= (return . ab)
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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.
   
== Special notation ==
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== '''<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))))
+
>>= (\_ -> 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 by the compiler to ordinary expressions that use <hask>Monad</hask> class functions.
 
   
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 the <hask>do</hask>-notation can be found in a section of [[Monads as computation#Do notation|Monads as computation]] and in other [[Monad#Monad tutorials|tutorials]].
+
More on <hask>do</hask>-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 ==
 
== Commutative monads ==
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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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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://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf 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/browse/code/monads/JavaMonads/ Java]
 
* [http://www.ccs.neu.edu/home/dherman/browse/code/monads/JavaMonads/ Java]
 
* [http://permalink.gmane.org/gmane.comp.lang.concatenative/1506 Joy]
 
* [http://permalink.gmane.org/gmane.comp.lang.concatenative/1506 Joy]
* [http://research.microsoft.com/~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)
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* [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/posts/a-monad-tutorial-for-ocaml/ 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.etsimo.uniovi.es/python/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]
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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://lambda-the-ultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambda-the-ultimate/ Lambda The Ultimate].
+
[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].
   
 
==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://hackage.haskell.org/packages/archive/mtl/1.1.0.2/doc/html/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 - memory-only effects]
+
* [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.html Global 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 effects]
+
* [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://hackage.haskell.org/packages/archive/mtl/latest/doc/html/Control-Monad-Error.html Functions which may 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://hackage.haskell.org/packages/archive/mtl/latest/doc/html/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 - unrestricted side effects]
+
* [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: computations with multiple choices]
+
* [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 computations]
 
* [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://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://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/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-exception Typed exceptions with call traces as a monad]
Line 203: Line 234:
 
* [http://hackage.haskell.org/package/monadiccp A constraint programming 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/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 instance of the monad abstraction out there. Please add them as you come across each species.
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==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==
 
==See also==
   
 
* [[What a Monad is not]]
 
* [[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|*]]
 
[[Category:Monad|*]]

Latest revision as of 20:30, 20 October 2019

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, 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 computation-produced 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 side-effects; etc..

Sometimes the specific monadic type also provides the ability to somehow 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 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 EDSL (embedded domain-specific 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 mind-twisting for the beginners, numerous tutorials that deal exclusively with monads were created (see monad tutorials).

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

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 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 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:

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

Fun

See also