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, | + | '''''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. |
− | Each monad, or computation type, provides means, subject to '''''Monad Laws''''', | + | 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 – 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) – 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. |
− | + | [[Image:Monads inter-dependencies 2.png|center]] | |
− | 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]]). | + | 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 == | == 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 | + | * 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 | ||
Line 37: | Line 39: | ||
</haskell> | </haskell> | ||
− | See [[Monad laws|this intuitive explanation]] of why they should obey the Monad laws. | + | 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 | Any Monad can be made a [[Functor]] by defining | ||
Line 83: | Line 85: | ||
<haskell> | <haskell> | ||
do | do | ||
− | a <- | + | a <- actA |
− | b <- | + | b <- actB |
m a b | m a b | ||
</haskell> | </haskell> | ||
Line 90: | Line 92: | ||
<haskell> | <haskell> | ||
do | do | ||
− | b <- | + | b <- actB |
− | a <- | + | a <- actA |
m a b | m a b | ||
</haskell> | </haskell> | ||
Line 103: | Line 105: | ||
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 [[ | + | 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:// | + | * [http://www.reddit.com/r/programming/comments/1761q/monads_in_c_pt_ii/ C] |
− | + | ||
* [http://cml.cs.uchicago.edu/pages/cml.html CML.event] ? | * [http://cml.cs.uchicago.edu/pages/cml.html CML.event] ? | ||
− | * [http:// | + | * [http://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf Clean] State monad |
− | + | ||
* [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 | + | * [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/ | + | * [http://research.microsoft.com/en-us/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ] |
− | * [http:// | + | * [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.cas.mcmaster.ca/~carette/metamonads/ MetaOcaml] | ** [http://www.cas.mcmaster.ca/~carette/metamonads/ MetaOcaml] | ||
− | ** [http://enfranchisedmind.com | + | ** [http://blog.enfranchisedmind.com/2007/08/a-monad-tutorial-for-ocaml/ A Monad Tutorial for Ocaml] |
− | * [http:// | + | * [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] | * [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Prolog] | ||
* Python | * Python | ||
− | ** [http:// | + | ** [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] | |
− | ** Twisted's [http:// | + | |
* 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. | + | ** [http://meta-meta.blogspot.com/2006/12/monads-in-ruby-part-1-identity.html and other implementation] |
− | + | ||
− | + | ||
− | + | ||
* Scheme: | * Scheme: | ||
** [http://okmij.org/ftp/Scheme/monad-in-Scheme.html Scheme] | ** [http://okmij.org/ftp/Scheme/monad-in-Scheme.html Scheme] | ||
Line 157: | Line 151: | ||
Unfinished: | Unfinished: | ||
− | |||
* [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 | ||
Line 166: | Line 159: | ||
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/ | + | * [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/ | + | * [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/ | + | * [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/ | + | * [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 | + | * [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 | + | * [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/ | + | * [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/ | + | * [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:// | + | * [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/ | + | * [http://hackage.haskell.org/packages/archive/stream-monad/latest/doc/html/Control-Monad-Stream.html non-determinism] - interleave computations with suspension. |
− | * [http:// | + | * [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:// | + | * [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:// | + | * [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:// | + | * [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://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/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] | ||
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* [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. | ||
Line 217: | Line 211: | ||
* [[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|*]] |
Revision as of 18:39, 3 August 2013
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 or state, etc.
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 – 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) – 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.
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 tutorials).
Contents |
1 Common monads
Most common applications of monads include:
- Representing failure using monadMaybe
- Nondeterminism using monad to represent carrying multiple valuesList
- State using monadState
- Read-only environment using monadReader
- I/O using monadIO
2 Monad class
Monads can be viewed as a standard programming interface to various data or control structures, which is captured by theclass 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.
3 Special notation
In order to improve the look of code that uses monads Haskell provides a special syntactic sugar calledthing1 >>= (\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
do x <- thing1 y <- func1 x thing2 z <- func2 y return z
4 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:
- monadReader
- monadMaybe
5 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.
6 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.
7 Monad research
A collection of research papers about monads.
8 Monads in other languages
Implementations of monads in other languages.
- C
- CML.event ?
- Clean State monad
- JavaScript
- Java
- Joy
- LINQ
- Lisp
- Miranda
- OCaml:
- Perl6 ?
- Prolog
- Python
- Python
- Twisted's Deferred monad
- Ruby:
- Scheme:
- Tcl
- The Unix Shell
- More monads by Oleg
- CLL: a concurrent language based on a first-order intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
Unfinished:
- Parsing, Maybe and Error in Tcl
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.
9 Interesting monads
A list of monads for various evaluation strategies and games:
- Identity monad - the trivial monad.
- Optional results from computations - error checking without null.
- Random values - run code in an environment with access to a stream of random numbers.
- Read only variables - guarantee read-only access to values.
- Writable state - i.e. log to a state buffer
- A supply of unique values - useful for e.g. guids or unique variable names
- ST - memory-only, locally-encapsulated mutable variables. Safely embed mutable state inside pure functions.
- Global state - a scoped, mutable state.
- Undoable state effects - roll back state changes
- Function application - chains of function application.
- Functions which may error - track location and causes of errors.
- Atomic memory transactions - software transactional memory
- Continuations - computations which can be interrupted and resumed.
- IO - unrestricted side effects on the world
- Search monad - bfs and dfs search environments.
- non-determinism - interleave computations with suspension.
- stepwise computation - encode non-deterministic choices as stepwise deterministic ones
- Backtracking computations
- Region allocation effects
- LogicT - backtracking monad transformer with fair operations and pruning
- concurrent events and threads - refactor event and callback heavy programs into straight-line code via co-routines
- QIO - The Quantum computing monad
- Pi calculus - a monad for Pi-calculus style concurrent programming
- Commutable monads for parallel programming
- Simple, Fair and Terminating Backtracking Monad
- Typed exceptions with call traces as a monad
- Breadth first list monad
- Continuation-based queues as monads
- Typed network protocol monad
- Non-Determinism Monad for Level-Wise Search
- Transactional state monad
- A constraint programming monad
- A probability distribution monad
- Sets - Set computations
- HTTP - http connections as a monadic environment
- 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.
10 Fun
- If you are tired of monads, you can easily get rid of them.