Difference between revisions of "Monad"

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(Update Monad page to reflect fact that Applicative is a superclass of Monad.)
(fix link to erik meijer linq webpage)
 
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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}}
   
'''''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, etc.
 
 
== The <code>Monad</code> class ==
 
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 &ndash; 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) &ndash; 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.
 
 
<haskell style="background-color:#f8f1ab;border-radius:15px;border:2px solid #000000;padding:15px">
 
# Monad Laws in pseudo code
 
(a) reaction $ value ==> action_description
 
 
(c) action_description `bind` reaction ==> action_description
 
 
(b) run action_description ==> value
 
</haskell>
 
 
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 ==
 
Most common applications of monads include:
 
* Representing failure using <hask>Maybe</hask> monad
 
* Nondeterminism using <hask>List</hask> monad to represent carrying multiple values
 
* State using <hask>State</hask> monad
 
* Read-only environment using <hask>Reader</hask> monad
 
* I/O 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 the <hask>Monad</hask> class. All common monads are members of it:
+
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>
Line 35: Line 13:
 
(>>) :: 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 the following equations, or '''''Monad Laws''''':
+
In addition to implementing the class functions, all instances of <code>Monad</code> should satisfy 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. 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.
 
  +
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. See [[Functor hierarchy proposal]].
+
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]].
   
== Special notation ==
 
  +
If the <code>Monad</code> definitions are preferred, <code>Functor</code> and <code>Applicative</code> instances can be defined from them with:
   
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:
 
  +
<haskell>
  +
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>
  +
  +
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>
 
<haskell>
Line 62: Line 61:
   
 
<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>
   
This can also be 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>
   
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.
 
  +
(the curly braces and the semicolons are optional when the indentation rules are observed).
   
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.
 
 
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>).
   
It is possible to intermix the <hask>do</hask>-notation with regular notation.
 
 
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.
   
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]].
 
 
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 ==
 
== Commutative monads ==
'''Commutative monads''' are monads for which the order of actions makes no difference (they '''commute'''), that is when following code:
+
For monads which are ''commutative'' the order of actions makes no difference (i.e. they ''commute''), so the following code:
 
<haskell>
 
<haskell>
 
do
 
do
Line 104: Line 106:
 
</haskell>
 
</haskell>
   
Examples of commutative include:
+
Examples of commutative monads include:
* <hask>Reader</hask> monad
+
* <code>Reader</code> monad
* <hask>Maybe</hask> 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.
   
 
See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials.
 
See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials.
Line 116: Line 118:
 
== 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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* [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/en-us/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ]
+
* [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://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]
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* [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:
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* [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.org/ Lambda The Ultimate].
 
   
 
==Interesting monads==
 
==Interesting monads==
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* [http://hackage.haskell.org/package/monad-memo Memoization] - add memoization to code
 
* [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==
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[[Category:Monad|*]]
 
[[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