Difference between revisions of "Monad"

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* [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]
* [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]
* [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
* [http://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf Clean] State monad
* [http://clojure.googlegroups.com/web/monads.clj Clojure]
* [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/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/~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)
* [http://sleepingsquirrel.org/monads/monads.lisp Lisp]
* [http://sleepingsquirrel.org/monads/monads.lisp 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://www.pps.jussieu.fr/~beffara/darcs/pivm/caml-vm/monad.mli also]
** [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://enfranchisedmind.com/blog/posts/a-monad-tutorial-for-ocaml/ A Monad Tutorial for Ocaml]
* [http://sleepingsquirrel.org/monads/monads.html Perl]
* [http://sleepingsquirrel.org/monads/monads.html Perl]
* [http://programming.reddit.com/info/p66e/comments Perl6 ?]
* [http://programming.reddit.com/info/p66e/comments Perl6 ?]

Revision as of 22:13, 29 October 2011

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. This lends monads to supplementing pure calculations with features like I/O, common environment or state, and to preprocessing of computations (simplification, optimization 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.

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.

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.

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 through backtracking using List monad
  • 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.

Any Monad can be made a Functor by defining

fmap ab ma = ma >>= (return . ab)

However, the Functor class is not a superclass of the Monad class. See Functor hierarchy proposal.

Special notation

In order to improve the look of code that uses monads Haskell provides a special syntactic sugar called do-notation. For example, 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 be also written using the do-notation as follows:

  x <- thing1
  y <- func1 x
  z <- func2 y
  return z

Code written using the do-notation is transformed by the compiler to ordinary expressions that use Monad class functions.

When using the 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 the 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:

  a <- f x
  b <- g y
  m a b

is the same as:

  b <- g y
  a <- f x
  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 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.


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.


See also