Difference between revisions of "Monad"
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(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 timelinemonad tutorials]].'' 

+   

+  
{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=base}} 
{{Standard classMonadmodule=Control.Monadmoduledoc=ControlMonadpackage=base}} 

−  The '''Monad''' class is defined like this: 

+  
+  == The <code>Monad</code> class == 

+  
+  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> 

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 

</haskell> 
</haskell> 

−  All instances of Monad should obey: 

+  In addition to implementing the class functions, all instances of <code>Monad</code> should satisfy the following equations, or ''monad laws'': 

<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> 

−  +  For more information, including an intuitive explanation of why the monad laws should be satisfied, see [[Monad laws]]. 

−  Any Monad can be made a [[Functor]] by defining 

+  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: 

+  
+  <haskell> 

⚫  
+   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 

+  * Readonly 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> 

+  thing1 >>= (\x > func1 x >>= (\y > thing2 

+  >>= (\_ > func2 y >>= (\z > return z)))) 

+  </haskell> 

+  
+  which can be written more clearly by breaking it into several lines and omitting parentheses: 

+  
+  <haskell> 

+  thing1 >>= \x > 

+  func1 x >>= \y > 

+  thing2 >>= \_ > 

+  func2 y >>= \z > 

+  return z 

+  </haskell> 

+  
+  can also be written using <code>do</code>notation: 

+  
+  <haskell> 

+  do { 

+  x < thing1 ; 

+  y < func1 x ; 

+  thing2 ; 

+  z < func2 y ; 

+  return z 

+  } 

+  </haskell> 

+  
+  (the curly braces and the semicolons are optional when the indentation rules are observed). 

+  
+  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 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. 

+  
+  More on <code>do</code>notation can be found in a section of [[Monads as computation#Do notationMonads as computation]] and in other [[Monad#Monad tutorialstutorials]]. 

+  
+  == Commutative monads == 

+  For monads which are ''commutative'' the order of actions makes no difference (i.e. they ''commute''), so the following code: 

+  <haskell> 

+  do 

+  a < actA 

+  b < actB 

+  m a b 

+  </haskell> 

+  is the same as: 

<haskell> 
<haskell> 

⚫  
+  do 

+  b < actB 

+  a < actA 

+  m a b 

</haskell> 
</haskell> 

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

+  Examples of commutative monads include: 

+  * <code>Reader</code> monad 

+  * <code>Maybe</code> monad 

== Monad tutorials == 
== Monad tutorials == 

−  Monads are known for being 
+  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. 
−  * [[Monads as containers]], [[Monads as computation]] 

+  See the [[Monad tutorials timeline]] for a comprehensive list of monad tutorials. 

−  * [http://haskell.org/all_about_monads/html/index.html All About Monads] 

−  * [[Simple monad examples]] 

−  * [http://www.loria.fr/~kow/monads/index.html Of monads and space suits] 

−  * [http://sigfpe.blogspot.com/2006/08/youcouldhaveinventedmonadsand.html You could have invented monads] 

−  * [[Meet Bob The Monadic Lover]], or the slightly more serious [[The Monadic Way]] 

−  * Computational monads [http://programming.reddit.com/info/ox6s/comments/coxiv part 1] and [http://programming.reddit.com/info/ox6s/comments/coxoh part 2]. 

−  * [http://www.alpheccar.org/fr/posts/show/60 Three kind of monads] : sequencing, side effects or containers 

−  * [http://haskell.org/haskellwiki/Books_and_tutorials#Using_monads More tutorials on monads] 

−  * [http://haskell.org/haskellwiki/Blog_articles#Monads Even more tutorials on monads..] 

== 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] 
+  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 HenkJan van Tuyl. 
== Monad research == 
== Monad research == 

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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] 
−  * [ 
+  * [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:// 
+  * [http://www.st.cs.ru.nl/papers/2010/CleanStdEnvAPI.pdf Clean] State monad 
* [http://cratylus.freewebspace.com/monadsinjavascript.htm JavaScript] 
* [http://cratylus.freewebspace.com/monadsinjavascript.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/ 
+  * [https://web.archive.org/web/20130522092554/http://research.microsoft.com/enus/um/people/emeijer/Papers/XLinq%20XML%20Programming%20Refactored%20(The%20Return%20Of%20The%20Monoids).htm LINQ] 
−  * [http:// 
+  * [http://commonlisp.net/project/clmonadmacros/monadmacros.htm Lisp] 
* [http://lambdatheultimate.org/node/1136#comment12448 Miranda] 
* [http://lambdatheultimate.org/node/1136#comment12448 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/metaocamlusersl/2005March/000057.html more] 
** [https://mailman.rice.edu/pipermail/metaocamlusersl/2005March/000057.html more] 

−  ** [http://www.pps.jussieu.fr/~beffara/darcs/pivm/camlvm/monad.mli also] 

** [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/amonadtutorialforocaml/ A Monad Tutorial for Ocaml] 
−  * [http:// 
+  * [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:// 
+  ** [http://code.activestate.com/recipes/439361/ Python] 
−  ** [http://www. 
+  ** 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/monadsinruby/00introduction.html Ruby] 
** [http://moonbase.rydia.net/mental/writings/programming/monadsinruby/00introduction.html Ruby] 

−  ** [http://metameta.blogspot.com/2006/12/monadsinrubypart1identity. 
+  ** [http://metameta.blogspot.com/2006/12/monadsinrubypart1identity.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/monadinScheme.html Scheme] 
** [http://okmij.org/ftp/Scheme/monadinScheme.html Scheme] 

** [http://www.ccs.neu.edu/home/dherman/research/tutorials/monadsforschemers.txt also] 
** [http://www.ccs.neu.edu/home/dherman/research/tutorials/monadsforschemers.txt also] 

+  ** Monads & Do notation: [https://eltramo.be/blog/asyncmonad/ Part 1] [https://eltramo.be/blog/schememonads/ 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/monadicshell.html The Unix Shell] 
* [http://okmij.org/ftp/Computation/monadicshell.html The Unix Shell] 

* [http://okmij.org/ftp/Computation/monads.html More monads by Oleg] 
* [http://okmij.org/ftp/Computation/monads.html More monads by Oleg] 

* [http://lambdatheultimate.org/node/2322 CLL]: a concurrent language based on a firstorder intuitionistic linear logic where all right synchronous connectives are restricted to a monad. 
* [http://lambdatheultimate.org/node/2322 CLL]: a concurrent language based on a firstorder intuitionistic linear logic where all right synchronous connectives are restricted to a monad. 

+  * [http://lambdatheultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambdatheultimate.org/ Lambda The Ultimate]. 

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 

−  And possibly there 
+  And possibly there exists: 
* Standard ML (via modules?) 
* Standard ML (via modules?) 

−  +  ''(If you know of other implementations, please add them here.)'' 

−  
−  [http://lambdatheultimate.org/node/1136 Collection of links to monad implementations in various languages.] on [http://lambdatheultimate/ Lambda The Ultimate]. 

==Interesting monads== 
==Interesting monads== 

Line 101:  Line 176:  
A list of monads for various evaluation strategies and games: 
A list of monads for various evaluation strategies and games: 

−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadIdentity.html Identity monad]  the trivial monad. 
−  * [http://haskell.org/ghc/docs/latest/html/libraries/base/DataMaybe.html Optional results] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/DataMaybe.html Optional results from computations]  error checking without null. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/monadmersennerandom/latest/doc/html/ControlMonadMersenneRandom.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/ControlMonadReader.html Read only variables]  guarantee readonly access to values. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadWriterLazy.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/ControlMonadST.html ST] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/ControlMonadST.html ST  memoryonly, locallyencapsulated mutable variables]. Safely embed mutable state inside pure functions. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadStateLazy.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/ControlMonadInstances.html Function application] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/ControlMonadInstances.html#t:Monad Function application]  chains of function application. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadError.html Functions which may error]  track location and causes of errors. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/stm/latest/doc/html/ControlMonadSTM.html Atomic memory transactions]  software transactional memory 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/mtl/latest/doc/html/ControlMonadCont.html Continuations]  computations which can be interrupted and resumed. 
−  * [http://haskell.org/ghc/docs/latest/html/libraries/base/SystemIO.html#t%3AIO IO] 
+  * [http://www.haskell.org/ghc/docs/latest/html/libraries/base/SystemIO.html#t%3AIO IO]  unrestricted side effects on the world 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/levelmonad/0.4.1/doc/html/ControlMonadLevels.html Search monad]  bfs and dfs search environments. 
−  * [http://haskell.org/ 
+  * [http://hackage.haskell.org/packages/archive/streammonad/latest/doc/html/ControlMonadStream.html nondeterminism]  interleave computations with suspension. 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/stepwise/latest/doc/html/ControlMonadStepwise.html stepwise computation]  encode nondeterministic choices as stepwise deterministic ones 
−  * [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Backtracking] 
+  * [http://logic.csci.unt.edu/tarau/research/PapersHTML/monadic.html Backtracking computations] 
−  * [http://www.cs.cornell.edu/people/fluet/research/rgnmonad/index.html Region allocation] 
+  * [http://www.cs.cornell.edu/people/fluet/research/rgnmonad/index.html Region allocation effects] 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/logict/0.5.0.2/doc/html/ControlMonadLogic.html LogicT]  backtracking monad transformer with fair operations and pruning 
−  * [http:// 
+  * [http://hackage.haskell.org/packages/archive/monadtask/latest/doc/html/ControlMonadTask.html concurrent events and threads]  refactor event and callback heavy programs into straightline code via coroutines 
−  * [http:// 
+  * [http://hackage.haskell.org/package/QIO QIO]  The Quantum computing monad 
−  * 
+  * [http://hackage.haskell.org/packages/archive/fullsessions/latest/doc/html/ControlConcurrentFullSession.html Pi calculus]  a monad for Picalculus style concurrent programming 
+  * [http://wwwfp.dcs.stand.ac.uk/~kh/papers/pasco94/subsubsectionstar3_3_2_3.html Commutable monads for parallel programming] 

+  * [http://hackage.haskell.org/package/streammonad Simple, Fair and Terminating Backtracking Monad] 

+  * [http://hackage.haskell.org/package/controlmonadexception Typed exceptions with call traces as a monad] 

+  * [http://hackage.haskell.org/package/controlmonadomega Breadth first list monad] 

+  * [http://hackage.haskell.org/package/controlmonadqueue Continuationbased queues as monads] 

+  * [http://hackage.haskell.org/package/fullsessions Typed network protocol monad] 

+  * [http://hackage.haskell.org/package/levelmonad NonDeterminism Monad for LevelWise Search] 

+  * [http://hackage.haskell.org/package/monadtx Transactional state 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/setmonad Sets]  Set computations 

+  * [http://hackage.haskell.org/package/httpmonad/ HTTP]  http connections as a monadic environment 

+  * [http://hackage.haskell.org/package/monadmemo Memoization]  add memoization to code 

−  There are many more interesting 
+  There are many more interesting instances of the monad abstraction out there. Please add them as you come across each species. 
==Fun== 
==Fun== 

−  * If you are tired of monads, you can easily [http:// 
+  * 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== 

+  
+  * [[What a Monad is not]] 

+  * [[Monads as containers]] 

+  * [[Monads as computation]] 

+  * [[Monad/ST]] 

+  * [http://www.haskellforall.com/2012/06/youcouldhaveinventedfreemonads.html Why free monads matter] (blog article) 

−  +  [[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.
import Control.Monad 
Contents
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  Readonly 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 HenkJan van Tuyl.
Monad research
A collection of research papers about monads.
Monads in other languages
Implementations of monads in other languages.
 C
 Clojure
 CML.event ?
 Clean State monad
 JavaScript
 Java
 Joy
 LINQ
 Lisp
 Miranda
 OCaml:
 Perl6 ?
 Prolog
 Python
 Python
 Twisted's Deferred monad
 Ruby:
 Scheme:
 Swift
 Tcl
 The Unix Shell
 More monads by Oleg
 CLL: a concurrent language based on a firstorder intuitionistic linear logic where all right synchronous connectives are restricted to a monad.
 Collection of links to monad implementations in various languages. on Lambda The Ultimate.
Unfinished:
 Parsing, Maybe and Error in Tcl
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:
 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 readonly 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  memoryonly, locallyencapsulated 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.
 nondeterminism  interleave computations with suspension.
 stepwise computation  encode nondeterministic 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 straightline code via coroutines
 QIO  The Quantum computing monad
 Pi calculus  a monad for Picalculus 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
 Continuationbased queues as monads
 Typed network protocol monad
 NonDeterminism Monad for LevelWise 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 instances of the monad abstraction out there. Please add them as you come across each species.
Fun
 If you are tired of monads, you can easily get rid of them.