Difference between revisions of "Monad"

From HaskellWiki
Jump to: navigation, search
(Undoing revision 64026 (made redundant by pre-existing 'Monad tutorials' section))
(Various textual changes)
Line 6: Line 6:
 
== The <code>Monad</code> class ==
 
== The <code>Monad</code> class ==
   
Monads can be viewed as a standard programming interface to various data or control structures, which is captured by the <code>Monad</code> 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 26: Line 26:
 
For more information, including an intuitive explanation of why they should be obeyed, see [[Monad laws]].
 
For more information, including an intuitive explanation of why they should be obeyed, 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. 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]].
   
If the <code>Monad</code> definitions are preferred, <code>Functor</code> and <code>Applicative</code> instances can be defined from them with
+
If the <code>Monad</code> definitions are preferred, <code>Functor</code> and <code>Applicative</code> instances can be defined from them with:
   
 
<haskell>
 
<haskell>
Line 40: Line 40:
 
</haskell>
 
</haskell>
   
although the recommended order is to define <code>return</code> as <code>pure</code>, if the two are the same.
+
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 ==
 
== Common monads ==
Most common applications of monads include:
 
  +
These include:
 
* Representing failure using <code>Maybe</code> monad
 
* Representing failure using <code>Maybe</code> monad
 
* Nondeterminism using <code>List</code> monad to represent carrying multiple values
 
* Nondeterminism using <code>List</code> monad to represent carrying multiple values
Line 52: Line 52:
 
== '''<code>do</code>'''-notation ==
 
== '''<code>do</code>'''-notation ==
   
In order to improve the look of code that uses monads Haskell provides a special [[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 form of [[syntactic sugar]] called <code>do</code>-notation. For example, the following expression:
   
 
<haskell>
 
<haskell>
Line 69: Line 69:
 
</haskell>
 
</haskell>
   
This can also be written using the <code>do</code>-notation as follows:
+
can also be written using <code>do</code>-notation:
   
 
<haskell>
 
<haskell>
Line 85: Line 85:
 
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>).
 
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 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.
+
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.
 
It is possible to intermix the <code>do</code>-notation with regular notation.
Line 166: Line 166:
 
* [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?)

Revision as of 02:55, 16 March 2021

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

For more information, including an intuitive explanation of why they should be obeyed, 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

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 exists:

  • 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