Monad in plain English For the OOP programmer with no FP background

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In terms that an OOP programmer would understand  without any functional programming background   what is a monad  What problem does it solve and what are the most common places it s used  Update To clarify the kind of understanding I was looking for  let   s say you were converting an FP application that had monads into an OOP application  What would you do to port the responsibilities of the monads to the OOP app

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You have a recent presentation  Monadologie -- professional help on type anxiety  by Christopher League  July 12th  2010   which is quite interesting on topics of continuation and monad  The video going with this  slideshare  presentation is actually available at vimeo  The Monad part start around 37 minutes in  on this one hour video  and starts with slide 42 of its 58 slide presentation   It is presented as  the leading design pattern for functional programming   but the language used in the examples is Scala  which is both OOP and functional  You can read more on Monad in Scala in the blog post  Monads - Another way to abstract computations in Scala   from Debasish Ghosh  March 27  2008       A type constructor M is a monad if it supports these operations      the return function def unit A   x  A   M A     called  bind  in Haskell  def flatMap A B   m  M A    f  A   gt  M B    M B     Other two can be written in term of the first two   def map A B   m  M A    f  A   gt  B   M B      flatMap m   x   gt  unit f x      def andThen A B   ma  M A    mb  M B    M B      flatMap ma   x   gt  mb     So for instance  in Scala     Option is a monad        def unit A   x  A   Option A    Some x       def flatMap A B  m Option A   f A   Option B    Option B          m match          case None    None        case Some x     f x             List is Monad        def unit A   x  A   List A    List x       def flatMap A B  m List A   f A   List B    List B          m match           case Nil    Nil         case x  xs    f x      flatMap xs  f            Monad are a big deal in Scala because of convenient syntax built to take advantage of Monad structures   for comprehension in Scala   for     i  lt - 1 to 4   j  lt - 1 to i   k  lt - 1 to j   yield i j k   is translated by the compiler to    1 to 4  flatMap   i   gt     1 to i  flatMap   j   gt       1 to j  map   k   gt        i j k       The key abstraction is the flatMap  which binds the computation through chaining  Each invocation of flatMap returns the same data structure type  but of different value   that serves as the input to the next command in chain   In the above snippet  flatMap takes as input a closure  SomeType    gt  List AnotherType  and returns a List AnotherType   The important point to note is that all flatMaps take the same closure type as input and return the same type as output   This is what  binds  the computation thread - every item of the sequence in the for-comprehension has to honor this same type constraint     If you take two operations  that may fail  and pass the result to the third  like   lookupVenue  String   gt  Option Venue  getLoggedInUser  SessionID   gt  Option User  reserveTable   Venue  User    gt  Option ConfNo    but without taking advantage of Monad  you get convoluted OOP-code like   val user   getLoggedInUser session  val confirm     if  user isDefined  None   else lookupVenue name  match       case None   gt  None     case Some venue    gt        val confno   reserveTable venue  user get        if confno isDefined          mailTo confno get  user get        confno       whereas with Monad  you can work with the actual types  Venue  User  like all the operations work  and keep the Option verification stuff hidden  all because of the flatmaps of the for syntax   val confirm   for     venue  lt - lookupVenue name    user  lt - getLoggedInUser session    confno  lt - reserveTable venue  user    yield     mailTo confno  user    confno     The yield part will only be executed if all three functions have Some X   any None would directly be returned to confirm      So      Monads allow ordered computation within Functional Programing  that allows us to model sequencing of actions in a nice structured form  somewhat like a DSL       And the greatest power comes with the ability to compose monads that serve different purposes  into extensible abstractions within an application       This sequencing and threading of actions by a monad is done by the language compiler that does the transformation through the magic of closures      By the way  Monad is not only model of computation used in FP      Category theory proposes many models of computation  Among them         the Arrow model of computations   the Monad model of computations   the Applicative model of computations

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I am sharing my understanding of Monads  which may not be theoretically perfect  Monads are about Context propagation  Monad is  you define some context for some data  or data type s    and then define how that context will be carried with the data throughout its processing pipeline  And defining context propagation is mostly about defining how to merge multiple contexts  of same type   Using Monads also means ensuring these contexts are not accidentally stripped off from the data  On the other hand  other context-less data can be brought into a new or existing context  Then this simple concept can be used to ensure compile time correctness of a program

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I ll try to make the shortest definition I can manage using OOP terms   A generic class CMonadic lt T gt  is a monad if it defines at least the following methods   class CMonadic lt T gt         static CMonadic lt T gt  create T t       a k a    return  in Haskell     public CMonadic lt U gt  flatMap lt U gt  Func lt T  CMonadic lt U gt  gt  f      a k a   bind  in Haskell     and if the following laws apply for all types T and their possible values t  left identity   CMonadic lt T gt  create t  flatMap f     f t    right identity  instance flatMap CMonadic lt T gt  create     instance   associativity   instance flatMap f  flatMap g     instance flatMap t   gt  f t  flatMap g     Examples    A List monad may have   List lt int gt  create 1  -- gt   1    And flatMap on the list  1 2 3  could work like so   intList flatMap x   gt  List lt int gt  makeFromTwoItems x  x 10   -- gt   1 10 2 20 3 30    Iterables and Observables can also be made monadic  as well as Promises and Tasks   Commentary   Monads are not that complicated  The flatMap function is a lot like the more commonly encountered map  It receives a function argument  also known as delegate   which it may call  immediately or later  zero or more times  with a value coming from the generic class  It expects that passed function to also wrap its return value in the same kind of generic class  To help with that  it provides create  a constructor that can create an instance of that generic class from a value  The return result of flatMap is also a generic class of the same type  often packing the same values that were contained in the return results of one or more applications of flatMap to the previously contained values  This allows you to chain flatMap as much as you want   intList flatMap x   gt  List lt int gt  makeFromTwo x  x 10           flatMap x   gt  x   3    0                       List lt string gt  create  x       x toString                          List lt string gt  empty      It just so happens that this kind of generic class is useful as a base model for a huge number of things  This  together with the category theory jargonisms  is the reason why Monads seem so hard to understand or explain  They re a very abstract thing and only become obviously useful once they re specialized   For example  you can model exceptions using monadic containers  Each container will either contain the result of the operation or the error that has occured  The next function  delegate  in the chain of flatMap callbacks will only be called if the previous one packed a value in the container  Otherwise if an error was packed  the error will continue to propagate through the chained containers until a container is found that has an error handler function attached via a method called  orElse    such a method would be an allowed extension   Notes  Functional languages allow you to write functions that can operate on any kind of a monadic generic class  For this to work  one would have to write a generic interface for monads  I don t know if its possible to write such an interface in C   but as far as I know it isn t    interface IMonad lt T gt         static IMonad lt T gt  create T t      not allowed     public IMonad lt U gt  flatMap lt U gt  Func lt T  IMonad lt U gt  gt  f      not specific enough         because the function must return the same kind of monad  not just any monad

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A simple Monads explanation with a Marvel s case study is here   Monads are abstractions used to sequence dependent functions that are effectful  Effectful here means they return a type in form F A  for example Option A  where Option is F  called type constructor  Let s see this in 2 simple steps   Below Function composition is transitive  So to go from A to C I can compose A    B and B    C     A   gt  C       A   gt  B  andThen  B   gt  C       However  if the function returns an effect type like Option A  i e  A    F B  the composition doesn t work as to go to B we need A    B but we have A    F B      We need a special operator   bind  that knows how to fuse these functions that return F A         A   gt  F C        A   gt  F B   bind  B   gt  F C    The  bind  function is defined for the specific F    There is also  return   of type A    F A  for any A  defined for that specific F also  To be a Monad  F must have these two functions defined for it   Thus we can construct an effectful function A    F B  from any pure function A    B     A   gt  F B        A   gt  B  andThen  return   but a given F can also define its own opaque  built-in  special functions of such types that a user can t define themself  in a pure language   like     random   Range    Random Int    print   String    IO         try     catch   etc

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In terms that an OOP programmer would   understand  without any functional   programming background   what is a   monad       What problem does it solve and what   are the most common places it s used are the most common places it s used    In terms of OO programming  a monad is an interface  or more likely a mixin   parameterized by a type  with two methods  return and bind that describe    How to inject a value to get a monadic value of that injected value type  How to use a function that makes a monadic value from a non-monadic one  on a monadic value    The problem it solves is the same type of problem you d expect from any interface  namely   I have a bunch of different classes that do different things  but seem to do those different things in a way that has an underlying similarity  How can I describe that similarity between them  even if the classes themselves aren t really subtypes of anything closer than  the Object  class itself    More specifically  the Monad  interface  is similar to IEnumerator or IIterator in that it takes a type that itself takes a type  The main  point  of Monad though is being able to connect operations based on the interior type  even to the point of having a new  internal type   while keeping - or even enhancing - the information structure of the main class

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UPDATE  This question was the subject of an immensely long blog series  which you can read at Monads     thanks for the great question      In terms that an OOP programmer would understand  without any functional programming background   what is a monad    A monad is an  amplifier  of types that obeys certain rules and which has certain operations provided   First  what is an  amplifier of types    By that I mean some system which lets you take a type and turn it into a more special type  For example  in C  consider Nullable lt T gt   This is an amplifier of types  It lets you take a type  say int  and add a new capability to that type  namely  that now it can be null when it couldn t before   As a second example  consider IEnumerable lt T gt   It is an amplifier of types  It lets you take a type  say  string  and add a new capability to that type  namely  that you can now make a sequence of strings out of any number of single strings   What are the  certain rules   Briefly  that there is a sensible way for functions on the underlying type to work on the amplified type such that they follow the normal rules of functional composition  For example  if you have a function on integers  say  int M int x    return x   N x   2       then the corresponding function on Nullable lt int gt  can make all the operators and calls in there work together  in the same way  that they did before    That is incredibly vague and imprecise  you asked for an explanation that didn t assume anything about knowledge of functional composition    What are the  operations      There is a  unit  operation  confusingly sometimes called the  return  operation  that takes a value from a plain type and creates the equivalent monadic value  This  in essence  provides a way to take a value of an unamplified type and turn it into a value of the amplified type  It could be implemented as a constructor in an OO language  There is a  bind  operation that takes a monadic value and a function that can transform the value  and returns a new monadic value  Bind is the key operation that defines the semantics of the monad  It lets us transform operations on the unamplified type into operations on the amplified type  that obeys the rules of functional composition mentioned before  There is often a way to get the unamplified type back out of the amplified type  Strictly speaking this operation is not required to have a monad   Though it is necessary if you want to have a comonad  We won t consider those further in this article     Again  take Nullable lt T gt  as an example  You can turn an int into a Nullable lt int gt  with the constructor  The C  compiler takes care of most nullable  lifting  for you  but if it didn t  the lifting transformation is straightforward  an operation  say    int M int x    whatever     is transformed into  Nullable lt int gt  M Nullable lt int gt  x          if  x    null           return null       else          return new Nullable lt int gt  whatever       And turning a Nullable lt int gt  back into an int is done with the Value property   It s the function transformation that is the key bit  Notice how the actual semantics of the nullable operation     that an operation on a null propagates the null     is captured in the transformation  We can generalize this    Suppose you have a function from int to int  like our original M   You can easily make that into a function that takes an int and returns a Nullable lt int gt  because you can just run the result through the nullable constructor  Now suppose you have this higher-order method   static Nullable lt T gt  Bind lt T gt  Nullable lt T gt  amplified  Func lt T  Nullable lt T gt  gt  func        if  amplified    null           return null      else         return func amplified Value       See what you can do with that  Any method that takes an int and returns an int  or takes an int and returns a Nullable lt int gt  can now have the nullable semantics applied to it   Furthermore  suppose you have two methods   Nullable lt int gt  X int q          Nullable lt int gt  Y int r            and you want to compose them   Nullable lt int gt  Z int s    return X Y s        That is  Z is the composition of X and Y  But you cannot do that because X takes an int  and Y returns a Nullable lt int gt   But since you have the  bind  operation  you can make this work   Nullable lt int gt  Z int s    return Bind Y s   X       The bind operation on a monad is what makes composition of functions on amplified types work  The  rules  I handwaved about above are that the monad preserves the rules of normal function composition  that composing with identity functions results in the original function  that composition is associative  and so on    In C    Bind  is called  SelectMany   Take a look at how it works on the sequence monad  We need to have two things  turn a value into a sequence and bind operations on sequences  As a bonus  we also have  turn a sequence back into a value   Those operations are   static IEnumerable lt T gt  MakeSequence lt T gt  T item        yield return item       Extract a value static T First lt T gt  IEnumerable lt T gt  sequence           let s just take the first one     foreach T item in sequence  return item       throw new Exception  No first item          Bind  is called  SelectMany  static IEnumerable lt T gt  SelectMany lt T gt  IEnumerable lt T gt  seq  Func lt T  IEnumerable lt T gt  gt  func        foreach T item in seq          foreach T result in func item               yield return result                  The nullable monad rule was  to combine two functions that produce nullables together  check to see if the inner one results in null  if it does  produce null  if it does not  then call the outer one with the result   That s the desired semantics of nullable     The sequence monad rule is  to combine two functions that produce sequences together  apply the outer function to every element produced by the inner function  and then concatenate all the resulting sequences together    The fundamental semantics of the monads are captured in the Bind SelectMany methods  this is the method that tells you what the monad really means   We can do even better  Suppose you have a sequences of ints  and a method that takes ints and results in sequences of strings  We could generalize the binding operation to allow composition of functions that take and return different amplified types  so long as the inputs of one match the outputs of the other   static IEnumerable lt U gt  SelectMany lt T U gt  IEnumerable lt T gt  seq  Func lt T  IEnumerable lt U gt  gt  func        foreach T item in seq          foreach U result in func item               yield return result                  So now we can say  amplify this bunch of individual integers into a sequence of integers  Transform this particular integer into a bunch of strings  amplified to a sequence of strings  Now put both operations together  amplify this bunch of integers into the concatenation of all the sequences of strings   Monads allow you to compose your amplifications      What problem does it solve and what are the most common places it s used    That s rather like asking  what problems does the singleton pattern solve    but I ll give it a shot    Monads are typically used to solve problems like    I need to make new capabilities for this type and still combine old functions on this type to use the new capabilities  I need to capture a bunch of operations on types and represent those operations as composable objects  building up larger and larger compositions until I have just the right series of operations represented  and then I need to start getting results out of the thing I need to represent side-effecting operations cleanly in a language that hates side effects   C  uses monads in its design  As already mentioned  the nullable pattern is highly akin to the  maybe monad   LINQ is entirely built out of monads  the SelectMany method is what does the semantic work of composition of operations   Erik Meijer is fond of pointing out that every LINQ function could actually be implemented by SelectMany  everything else is just a convenience       To clarify the kind of understanding I was looking for  let s say you were converting an FP application that had monads into an OOP application  What would you do to port the responsibilities of the monads into the OOP app    Most OOP languages do not have a rich enough type system to represent the monad pattern itself directly  you need a type system that supports types that are higher types than generic types  So I wouldn t try to do that  Rather  I would implement generic types that represent each monad  and implement methods that represent the three operations you need  turning a value into an amplified value   maybe  turning an amplified value into a value  and transforming a function on unamplified values into a function on amplified values   A good place to start is how we implemented LINQ in C   Study the SelectMany method  it is the key to understanding how the sequence monad works in C   It is a very simple method  but very powerful     Suggested  further reading    For a more in-depth and theoretically sound explanation of monads in C   I highly recommend my  Eric Lippert s  colleague Wes Dyer s article on the subject  This article is what explained monads to me when they finally  clicked  for me    The Marvels of Monads  A good illustration of why you might want a monad around  uses Haskell in it s examples     You Could Have Invented Monads   And Maybe You Already Have   by Dan Piponi  Sort of   translation  of the previous article to JavaScript    Translation from Haskell to JavaScript of selected portions of the best introduction to monads I   ve ever read by James Coglan

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A monad is a data type that encapsulates a value  and to which  essentially  two operations can be applied    return x creates a value of the monad type that encapsulates x m  gt  gt   f  read it as  the bind operator   applies the function f to the value in the monad m   That s what a monad is  There are a few more technicalities  but basically those two operations define a monad  The real question is   What a monad does    and that depends on the monad     lists are monads  Maybes are monads  IO operations are monads  All that it means when we say those things are monads is that they have the monad interface of return and  gt  gt

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To respect fast readers  I start with precise definition first  continue with quick more  plain English  explanation  and then move to examples   Here is a both concise and precise definition slightly reworded      A monad  in computer science  is formally a map that          sends every type X of some given programming language to a new type T X   called the  type of T-computations with values in X      equipped with a rule for composing two functions of the form    f X- gt T Y  and g Y- gt T Z  to a function g  f X- gt T Z     in a way that is associative in the evident sense and unital with respect to a given unit function called pure X X- gt T X   to be thought of as taking a value to the pure computation that simply returns that value       So in simple words  a monad is a rule to pass from any type X to another type T X   and a rule to pass from two functions f X- gt T Y  and g Y- gt T Z   that you would like to compose but can t  to a new function h X- gt T Z   Which  however  is not the composition in strict mathematical sense  We are basically  bending  function s composition or re-defining how functions are composed    Plus  we require the monad s rule of composing to satisfy the  obvious  mathematical axioms    Associativity  Composing f with g and then with h  from outside  should be the same as composing g with h and then with f  from inside   Unital property  Composing f with the identity function on either side should yield f    Again  in simple words  we can t just go crazy re-defining our function composition as we like     We first need the associativity to be able to compose several functions in a row e g  f g h k x     and not to worry about specifying the order composing function pairs  As the monad rule only prescribes how to compose a pair of functions  without that axiom  we would need to know which pair is composed first and so on   Note that is different from the commutativity property that f composed with g were the same as g composed with f  which is not required    And second  we need the unital property  which is simply to say that identities compose trivially the way we expect them  So we can safely refactor functions whenever those identities can be extracted    So again in brief  A monad is the rule of type extension and composing functions satisfying the two axioms -- associativity and unital property   In practical terms  you want the monad to be implemented for you by the language  compiler or framework that would take care of composing functions for you  So you can focus on writing your function s logic rather than worrying how their execution is implemented    That is essentially it  in a nutshell     Being professional mathematician  I prefer to avoid calling h the  composition  of f and g  Because mathematically  it isn t  Calling it the  composition  incorrectly presumes that h is the true mathematical composition  which it isn t  It is not even uniquely determined by f and g  Instead  it is the result of our monad s new  rule of  composing  the functions  Which can be totally different from the actual mathematical composition even if the latter exists     To make it less dry  let me try to illustrate it by example that I am annotating with small sections  so you can skip right to the point   Exception throwing as Monad examples  Suppose we want to compose two functions   f  x - gt  1   x g  y - gt  2   y   But f 0  is not defined  so an exception e is thrown  Then how can you define the compositional value g f 0    Throw an exception again  of course  Maybe the same e  Maybe a new updated exception e1   What precisely happens here  First  we need new exception value s    different or same   You can call them nothing or null or whatever but the essence remains the same -- they should be new values  e g  it should not be a number in our example here  I prefer not to call them null to avoid confusion with how null can be implemented in any specific language  Equally I prefer to avoid nothing because it is often associated with null  which  in principle  is what null should do  however  that principle often gets bended for whatever practical reasons   What is exception exactly   This is a trivial matter for any experienced programmer but I d like to drop few words just to extinguish any worm of confusion   Exception is an object encapsulating information about how the invalid result of execution occurred   This can range from throwing away any details and returning a single global value  like NaN or null  or generating a long log list or what exactly happened  send it to a database and replicating all over the distributed data storage layer     The important difference between these two extreme examples of exception is that in the first case there are no side-effects  In the second there are  Which brings us to the  thousand-dollar  question   Are exceptions allowed in pure functions   Shorter answer  Yes  but only when they don t lead to side-effects    Longer answer  To be pure  your function s output must be uniquely determined by its input  So we amend our function f by sending 0 to the new abstract value e that we call exception  We make sure that value e contains no outside information that is not uniquely determined by our input  which is x  So here is an example of exception without side-effect   e       type  error     message   I got error trying to divide 1 by 0      And here is one with side-effect   e       type  error     message   Our committee to decide what is 1 0 is currently away      Actually  it only has side-effects if that message can possibly change in the future  But if it is guaranteed to never change  that value becomes uniquely predictable  and so there is no side-effect   To make it even sillier  A function returning 42 ever is clearly pure  But if someone crazy decides to make 42 a variable that value might change  the very same function stops being pure under the new conditions   Note that I am using the object literal notation for simplicity to demonstrate the essence  Unfortunately things are messed-up in languages like JavaScript  where error is not a type that behaves the way we want here with respect to function composition  whereas actual types like null or NaN do not behave this way but rather go through the some artificial and not always intuitive type conversions   Type extension  As we want to vary the message inside our exception  we are really declaring a new type E for the whole exception object and then  That is what the maybe number does  apart from its confusing name  which is to be either of type number or of the new exception type E  so it is really the union number   E of number and E  In particular  it depends on how we want to construct E  which is neither suggested nor reflected in the name maybe number   What is functional composition   It is the mathematical operation taking functions f  X - gt  Y and g  Y - gt  Z and constructing  their composition as function h  X - gt  Z satisfying h x    g f x    The problem with this definition occurs when the result f x  is not allowed as argument of g    In mathematics those functions cannot be composed without extra work  The strictly mathematical solution for our above example of f and g is to remove 0 from the set of definition of f  With that new set of definition  new more restrictive type of x   f becomes composable with g   However  it is not very practical in programming to restrict the set of definition of f like that  Instead  exceptions can be used    Or as another approach  artificial values are created like NaN  undefined  null  Infinity etc  So you evaluate 1 0 to Infinity and 1 -0 to -Infinity  And then force the new value back into your expression instead of throwing exception  Leading to results you may or may not find predictable   1 0                     gt  Infinity parseInt Infinity       gt  NaN NaN  lt  0                 gt  false false   1               gt  1   And we are back to regular numbers ready to move on     JavaScript allows us to keep executing numerical expressions at any costs without throwing errors as in the above example  That means  it also allows to compose functions  Which is exactly what monad is about - it is a rule to compose functions satisfying the axioms as defined at the beginning of this answer   But is the rule of composing function  arising from JavaScript s implementation for dealing with numerical errors  a monad   To answer this question  all you need is to check the axioms  left as exercise as not part of the question here     Can throwing exception be used to construct a monad   Indeed  a more useful monad would instead be the rule prescribing that if f throws exception for some x  so does its composition with any g  Plus make the exception E globally unique with only one possible value ever  terminal object in category theory   Now the two axioms are instantly checkable and we get a very useful monad  And the result is what is well-known as the maybe monad

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From wikipedia      In functional programming  a monad  is   a kind of abstract data type used to   represent computations   instead of   data in the domain model   Monads   allow the programmer to chain actions   together to build a pipeline  in which   each action is decorated with   additional processing rules provided   by the monad  Programs written in   functional style can make use of   monads to structure procedures that   include sequenced operations 1 2     or to define arbitrary control flows    like handling concurrency    continuations  or exceptions        Formally  a monad is constructed by   defining two operations  bind and   return  and a type constructor M that   must fulfill several properties to   allow the correct composition of   monadic functions  i e  functions that   use values from the monad as their   arguments   The return operation takes   a value from a plain type and puts it   into a monadic container of type M    The bind operation performs the   reverse process  extracting the   original value from the container and   passing it to the associated next   function in the pipeline       A programmer will compose monadic   functions to define a data-processing   pipeline  The monad acts as a   framework  as it s a reusable behavior   that decides the order in which the   specific monadic functions in the   pipeline are called  and manages all   the undercover work required by the   computation  3  The bind and return   operators interleaved in the pipeline   will be executed after each monadic   function returns control  and will   take care of the particular aspects   handled by the monad    I believe it explains it very well

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Whether a monad has a  natural  interpretation in OO depends on the monad   In a language like Java  you can translate the maybe monad to the language of checking for null pointers  so that computations that fail  i e   produce Nothing in Haskell  emit null pointers as results   You can translate the state monad into the language generated by creating a mutable variable and methods to change its state   A monad is a monoid in the category of endofunctors   The information that sentence puts together is very deep   And you work in a monad with any imperative language   A monad is a  sequenced  domain specific language   It satisfies certain interesting properties  which taken together make a monad a mathematical model of  imperative programming    Haskell makes it easy to define small  or large  imperative languages  which can be combined in a variety of ways   As an OO programmer  you use your language s class hierarchy to organize the kinds of functions or procedures that can be called in a context  what you call an object   A monad is also an abstraction on this idea  insofar as different monads can be combined in arbitrary ways  effectively  importing  all of the sub-monad s methods into the scope     Architecturally  one then uses type signatures to explicitly express which contexts may be used for computing a value   One can use monad transformers for this purpose  and there is a high quality collection of all of the  standard  monads     Lists   non-deterministic computations  by treating a list as a domain  Maybe   computations that can fail  but for which reporting is unimportant  Error   computations that can fail and require exception handling Reader  computations that can be represented by compositions of plain Haskell functions  Writer  computations with sequential  rendering   logging   to strings  html etc  Cont    continuations  IO      computations that depend on the underlying computer system  State   computations whose context contains a modifiable value    with corresponding monad transformers and type classes   Type classes allow a complementary approach to combining monads by unifying their interfaces  so that concrete monads can implement a standard interface for the monad  kind    For example  the module Control Monad State contains a class MonadState s m  and  State s  is an instance of the form  instance MonadState s  State s  where     put           get         The long story is that a monad is a functor which attaches  context  to a value  which has a way to inject a value into the monad  and which has a way to evaluate values with respect to the context attached to it  at least in a restricted way   So   return    a - gt  m a   is a function which injects a value of type a into a monad  action  of type m a     gt  gt       m a - gt   a - gt  m b  - gt  m b   is a function which takes a monad action  evaluates its result  and applies a function to the result   The neat thing about       is that the result is in the same monad   In other words  in m     f        pulls the result out of m  and binds it to f  so that the result is in the monad    Alternatively  we can say that       pulls f into m and applies it to the result    As a consequence  if we have f    a -  m b  and g    b -  m c  we can  sequence  actions   m  gt  gt   f  gt  gt   g   Or  using  do notation   do x  lt - m    y  lt - f x    g y   The type for      might be illuminating   It is     gt  gt      m a - gt  m b - gt  m b   It corresponds to the     operator in procedural languages like C   It allows do notation like   m   do x  lt - someQuery        someAction x        theNextAction        andSoOn   In mathematical and philosopical logic  we have frames and models  which are  naturally  modelled with monadism   An interpretation is a function which looks into the model s domain and computes the truth value  or generalizations  of a proposition  or formula  under generalizations    In a modal logic for necessity  we might say that a proposition is necessary if it is true in  every possible world  -- if it is true with respect to every admissible domain   This means that a model in a language for a proposition can be reified as a model whose domain consists of collection of distinct models  one corresponding to each possible world    Every monad has a method named  join  which flattens layers  which implies that every monad action whose result is a monad action can be embedded in the monad    join    m  m a  - gt  m a   More importantly  it means that the monad is closed under the  layer stacking  operation   This is how monad transformers work   they combine monads by providing  join-like  methods for types like  newtype MaybeT m a   MaybeT   runMaybeT    m  Maybe a      so that we can transform an action in  MaybeT m  into an action in m  effectively collapsing layers   In this case  runMaybeT    MaybeT m a -  m  Maybe a  is our join-like method    MaybeT m  is a monad  and MaybeT    m  Maybe a  -  MaybeT m a is effectively a constructor for a new type of monad action in m     A free monad for a functor is the monad generated by stacking f  with the implication that every sequence of constructors for f is an element of the free monad  or  more exactly  something with the same shape as the tree of sequences of constructors for f    Free monads are a useful technique for constructing flexible monads with a minimal amount of boiler-plate   In a Haskell program  I might use free monads to define simple monads for  high level system programming  to help maintain type safety  I m just using types and their declarations   Implementations are straight-forward with the use of combinators    data RandomF r a   GetRandom  r - gt  a  deriving Functor type Random r a   Free  RandomF r  a   type RandomT m a   Random  m a   m a  -- model randomness in a monad by computing random monad elements  getRandom        Random r r runRandomIO      Random r a - gt  IO a  use some kind of IO-based backend to run  runRandomIO      Random r a - gt  IO a  use some other kind of IO-based backend  runRandomList    Random r a - gt   a    some kind of list-based backend  for pseudo-randoms     Monadism is the underlying architecture for what you might call the  interpreter  or  command  pattern  abstracted to its clearest form  since every monadic computation must be  run   at least trivially   The runtime system runs the IO monad for us  and is the entry point to any Haskell program   IO  drives  the rest of the computations  by running IO actions in order    The type for join is also where we get the statement that a monad is a monoid in the category of endofunctors   Join is typically more important for theoretical purposes  in virtue of its type   But understanding the type means understanding monads   Join and monad transformer s join-like types are effectively compositions of endofunctors  in the sense of function composition   To put it in a Haskell-like pseudo-language   Foo    m  m a   lt -   m   m  a

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Why do we need monads    We want to program only using functions    functional programming  after all -FP   Then  we have a first big problem  This is a program   f x    2   x  g x y    x   y  How can we say  what is to be executed first  How can we form an ordered sequence of functions  i e  a program  using no more than functions   Solution  compose functions  If you want first g and then f  just write f g x y    OK  but     More problems  some functions might fail  i e  g 2 0   divide by 0   We have no  exceptions  in FP  How do we solve it   Solution  Let s allow functions to return two kind of things  instead of having g   Real Real - gt  Real  function from two reals into a real   let s allow g   Real Real - gt  Real   Nothing  function from two reals into  real or nothing     But functions should  to be simpler  return only one thing    Solution  let s create a new type of data to be returned  a  boxing type  that encloses maybe a real or be simply nothing  Hence  we can have g   Real Real - gt  Maybe Real  OK  but     What happens now to f g x y    f is not ready to consume a Maybe Real  And  we don t want to change every function we could connect with g to consume a Maybe Real   Solution  let s have a special function to  connect   compose   link  functions  That way  we can  behind the scenes  adapt the output of one function to feed the following one    In our case   g  gt  gt   f  connect compose g to f   We want  gt  gt   to get g s output  inspect it and  in case it is Nothing just don t call f and return Nothing  or on the contrary  extract the boxed Real and feed f with it   This algorithm is just the implementation of  gt  gt   for the Maybe type   Many other problems arise which can be solved using this same pattern  1  Use a  box  to codify store different meanings values  and have functions like g that return those  boxed values   2  Have composers linkers g  gt  gt   f to help connecting g s output to f s input  so we don t have to change f at all  Remarkable problems that can be solved using this technique are     having a global state that every function in the sequence of functions   the program   can share  solution StateMonad   We don t like  impure functions   functions that yield different output for same input  Therefore  let s mark those functions  making them to return a tagged boxed value  IO monad     Total happiness

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I would say the closest OO analogy to monads is the  command pattern    In the command pattern you wrap an ordinary statement or expression in a command object  The command object expose an execute method which executes the wrapped statement  So statement are turned into first class objects which can passed around and executed at will  Commands can be composed so you can create a program-object by chaining and nesting command-objects    The commands are executed by a separate object  the invoker  The benefit of using the command pattern  rather than just execute a series of ordinary statements  is that different invokers can apply different logic to how the commands should be executed   The command pattern could be used to add  or remove  language features which is not supported by the host language  For example  in a hypothetical OO language without exceptions  you could add exception semantics by exposing  try  and  throw  methods to the commands  When a command calls throw  the invoker backtracks through the list  or tree  of commands until the last  try  call  Conversely  you could remove exception semantic from a language  if you believe exceptions are bad  by catching all exceptions thrown by each individual commands  and turning them into error codes which are then passed to the next command   Even more fancy execution semantics like transactions  non-deterministic execution or continuations can be implemented like this in a language which doesn t support it natively  It is a pretty powerful pattern if you think about it   Now in reality the command-patterns is not used as a general language feature like this  The overhead of turning each statement into a separate class would lead to an unbearable amount of boilerplate code  But in principle it can be used to solve the same problems as monads are used to solve in fp

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Monads in typical usage are the functional equivalent of procedural programming s exception handling mechanisms   In modern procedural languages  you put an exception handler around a sequence of statements  any of which may throw an exception  If any of the statements throws an exception  normal execution of the sequence of statements halts and transfers to an exception handler    Functional programming languages  however  philosophically avoid exception handling features due to the  goto  like nature of them  The functional programming perspective is that functions should not have  side-effects  like exceptions that disrupt program flow   In reality  side-effects cannot be ruled out in the real world due primarily to I O  Monads in functional programming are used to handle this by taking a set of chained function calls  any of which might produce an unexpected result  and turning any unexpected result into encapsulated data that can still flow safely through the remaining function calls   The flow of control is preserved but the unexpected event is safely encapsulated and handled

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A monad is an array of functions    Pst  an array of functions is just a computation    Actually  instead of a true array  one function in one cell array  you have those functions chained by another function      The     allows to adapt the results from function i to feed function i 1  perform calculations between them  or  even  not to call function i 1    The types used here are  types with context   This is  a value with a  tag   The functions being chained must take a  naked value  and return a tagged result  One of the duties of     is to extract a naked value out of its context  There is also the function  return   that takes a naked value and puts it with a tag   An example with Maybe  Let s use it to store a simple integer on which make calculations   -- a   b multiply    Int - gt  Int - gt  Maybe Int multiply a b   return   a b   -- divideBy 5 100   100   5 divideBy    Int - gt  Int - gt  Maybe Int divideBy 0     Nothing -- dividing by 0 gives NOTHING divideBy denom num   return  quot num denom  -- quotient of num   denom  -- tagged value val1   Just 160   -- array of functions feeded with val1 array1   val1  gt  gt   divideBy 2   gt  gt   multiply 3  gt  gt   divideBy  4  gt  gt   multiply 3  -- array of funcionts created with the do notation -- equals array1 but for the feeded val1 array2    Int - gt  Maybe Int array2 n   do        v  lt - divideBy 2  n        v  lt - multiply 3 v        v  lt - divideBy 4 v        v  lt - multiply 3 v        return v  -- array of functions   -- the first  gt  gt   performs 160   0  returning Nothing -- the second  gt  gt   has to perform Nothing  gt  gt   multiply 3      -- and simply returns Nothing without calling multiply 3      array3   val1  gt  gt   divideBy 0   gt  gt   multiply 3  gt  gt   divideBy  4  gt  gt   multiply 3  main   do      print array1      print  array2 160       print array3   Just to show that monads are array of functions with helper operations  consider the equivalent to the above example  just using a real array of functions  type MyMonad    Int - gt  Maybe Int  -- my monad as a real array of functions  myArray1    divideBy 2  multiply 3  divideBy 4  multiply 3   -- function for the machinery of executing each function i with the result provided by function i-1 runMyMonad    Maybe Int - gt  MyMonad - gt  Maybe Int runMyMonad val      val runMyMonad Nothing     Nothing runMyMonad  Just val   f fs    runMyMonad  f val  fs   And it would be used like this   print  runMyMonad  Just 160  myArray1

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See my answer to  What is a monad    It begins with a motivating example  works through the example  derives an example of a monad  and formally defines  monad    It assumes no knowledge of functional programming and it uses pseudocode with function argument     expression syntax with the simplest possible expressions   This C   program is an implementation of the pseudocode monad   For reference  M is the type constructor  feed is the  bind  operation  and wrap is the  return  operation     include  lt iostream gt   include  lt string gt   template  lt class A gt  class M   public      A val      std  string messages      template  lt class A  class B gt  M lt B gt  feed M lt B gt    f  A   M lt A gt  x        M lt B gt  m   f x val       m messages   x messages   m messages      return m     template  lt class A gt  M lt A gt  wrap A x        M lt A gt  m      m val   x      m messages           return m     class T     class U     class V      M lt U gt  g V x        M lt U gt  m      m messages    called g  n       return m     M lt T gt  f U x        M lt T gt  m      m messages    called f  n       return m     int main         V x      M lt T gt  m   feed f  feed g  wrap x         std  cout  lt  lt  m messages

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From a practical point of view  summarizing what has been said in many previous answers and related articles   it seems to me that one of the fundamental  purposes   or usefulness  of the monad is to leverage the dependencies implicit in recursive method invocations aka function composition  i e  when f1 calls f2 calls f3  f3 needs to be evaluated before f2 before f1  to represent sequential composition in a natural way  especially in the context of a lazy evaluation model  that is  sequential composition as a plain sequence  e g   f3    f2    f1     in C - the trick is especially obvious if you think of a case where f3  f2 and f1 actually return nothing  their chaining as f1 f2 f3   is artificial  purely intended to create sequence     This is especially relevant when side-effects are involved  i e  when some state is altered  if f1  f2  f3 had no side-effects  it wouldn t matter in what order they re evaluated  which is a great property of pure functional languages  to be able to parallelize those computations for example   The more pure functions  the better   I think from that narrow point of view  monads could be seen as syntactic sugar for languages that favor lazy evaluation  that evaluate things only when absolutely necessary  following an order that does not rely on the presentation of the code   and that have no other means of representing sequential composition  The net result is that sections of code that are  impure   i e  that do have side-effects  can be presented naturally  in an imperative manner  yet are cleanly separated from pure functions  with no side-effects   which can be evaluated lazily   This is only one aspect though  as warned here

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In OO terms  a monad is a fluent container   The minimum requirement is a definition of class  lt A gt  Something that supports a constructor Something A a  and at least one method Something lt B gt  flatMap Function lt A  Something lt B gt  gt    Arguably  it also counts if your monad class has any methods with signature Something lt B gt  work   which preserves the class s rules -- the compiler bakes in flatMap at compile time   Why is a monad useful  Because it is a container that allows chain-able operations that preserve semantics  For example  Optional lt   gt  preserves the semantics of isPresent for Optional lt String gt   Optional lt Integer gt   Optional lt MyClass gt   etc   As a rough example   Something lt Integer gt  i   new Something  a      flatMap doOneThing     flatMap doAnother     flatMap toInt    Note we start with a string and end with an integer  Pretty cool   In OO  it might take a little hand-waving  but any method on Something that returns another subclass of Something meets the criterion of a container function that returns a container of the original type   That s how you preserve semantics -- i e  the container s meaning and operations don t change  they just wrap and enhance the object inside the container

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The simplest explanation I can think of is that monads are a way of composing functions with embelished results  aka Kleisli composition   An  embelished  function has the signature a - gt   b  smth  where a and b are types  think Int  Bool  that might be different from each other  but not necessarily - and smth is the  context  or the  embelishment    This type of functions can also be written a - gt  m b where m is equivalent to the  embelishment  smth  So these are functions that return values in context  think functions that log their actions  where smth is the logging message  or functions that perform input output and their results depends on the result of the IO action    A monad is an interface   typeclass   that makes the implementer tell it how to compose such functions  The implementer needs to define a composition function  a - gt  m b  - gt   b - gt  m c  - gt   a - gt  m c  for any type m that wants to implement the interface  this is the Kleisli composition    So  if we say that we have a tuple type  Int  String  representing results of computations on Ints that also log their actions  with     String  being the  embelishment  - the log of the action - and two functions increment    Int - gt   Int  String  and twoTimes    Int - gt   Int  String  we want to obtain a function incrementThenDouble    Int - gt   Int  String  which is the composition of the two functions that also takes into account the logs   On the given example  a monad implementation of the two functions applies to integer value 2 incrementThenDouble 2  which is equal to twoTimes  increment 2   would return  6    Adding 1  Doubling 3    for intermediary results increment 2 equal to  3    Adding 1    and twoTimes 3 equal to  6    Doubling 3     From this Kleisli composition function one can derive the usual monadic functions

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If you ve ever used Powershell  the patterns Eric described should sound familiar   Powershell cmdlets are monads  functional composition is represented by a pipeline   Jeffrey Snover s interview with Erik Meijer goes into more detail

[oop] Monad in plain English? (For the OOP programmer with no FP background)

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