Pollard's p − 1 algorithm

In functional programming, a monad is a structure that represents computations defined as sequences of steps. A type with a monad structure defines what it means to chain operations, or nest functions of that type together. This allows the programmer to build pipelines that process data in steps, in which each action is decorated with additional processing rules provided by the monad. As such, monads have been described as "programmable semicolons"; a semicolon is the operator used to chain together individual statements in many imperative programming languages, thus the expression implies that extra code will be executed between the statements in the pipeline. Monads have also been explained with a physical metaphor as assembly lines, where a conveyor belt transports data between functional units that transform it one step at a time. They can also be seen as a functional design pattern to build generic types.

Purely functional programs can use monads to structure procedures that include sequenced operations like those found in structured programming. Many common programming concepts can be described in terms of a monad structure, including side effects such as input/output, variable assignment, exception handling, parsing, nondeterminism, concurrency, and continuations. This allows these concepts to be defined in a purely functional manner, without major extensions to the language's semantics. Languages like Haskell provide monads in the standard core, allowing programmers to reuse large parts of their formal definition and apply in many different libraries the same interfaces for combining functions.

Formally, a monad consists of a type constructor M and two operations, bind and return (where return is often also called unit). The operations must fulfill several properties to allow the correct composition of monadic functions (i.e. functions that use values from the monad as their arguments or return value). The return operation takes a value from a plain type and puts it into a monadic container using the constructor, creating a monadic value. 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, possibly with additional checks and transformations. Because a monad can insert additional operations around a program's domain logic, monads can be considered a sort of aspect-oriented programming. The domain logic can be defined by the application programmer in the pipeline, while required aside bookkeeping operations can be handled by a pre-defined monad built in advance.

The name and concept comes from the eponymous construct (monad) in category theory, where monads are one particular kind of functor, a mapping between categories; although the term monad in functional programming contexts is usually used with a meaning corresponding to that of the term strong monad in category theory.

History

The concept of monad programming appeared in the 1980s in the programming language Opal even though it was called "commands" and never formally specified.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Eugenio Moggi first described the general use of monads to structure programs in 1991. Several people built on his work, including programming language researchers Philip Wadler and Simon Peyton Jones (both of whom were involved in the specification of Haskell). Early versions of Haskell used a problematic "lazy list" model for I/O, and Haskell 1.3 introduced monads as a more flexible way to combine I/O with lazy evaluation.

In addition to I/O, programming language researchers and Haskell libraries designers have successfully applied monads to topics including parsers and programming language interpreters. The concept of monads along with the Haskell do-notation for them has also been generalized to form applicative functors and arrows.

For a long time, Haskell and its derivatives have been the only major users of monads in programming. There also exist formulations in Scheme, Perl, Python, Racket, Clojure and Scala, and monads have been an option in the design of a new ML standard. Recently F# has included a feature called computation expressions or workflows, which are an attempt to introduce monadic constructs within a syntax more palatable to programmers with an imperative background.

Motivating examples

The Haskell programming language is a functional language that makes heavy use of monads, and includes syntactic sugar to make monadic composition more convenient. All of the code samples in this article are written in Haskell unless noted otherwise.

We demonstrate two common examples given when introducing monads: the Maybe monad and the I/O monad. Monads are of course not restricted to the Haskell language, though: the second example shows the Writer monad in JavaScript.

The Maybe monad

Now consider the option type Maybe a, representing a value that is either a single value of type a, or no value at all. To distinguish these, we have two algebraic data type constructors: Just x, containing the value x, or Nothing, containing no value.

data Maybe t = Just t | Nothing

We would like to be able to use this type as a simple sort of checked exception: at any point in a computation, the computation may fail, which causes the rest of the computation to be skipped and the final result to be Nothing. If all steps of the calculation succeed, the final result is Just x for some value x.

In the following example, add is a function that takes two arguments of type Maybe Int, and returns a result of the same type. If both mx and my have Just values, we want to return Just their sum; but if either mx or my is Nothing, we want to return Nothing. If we naïvely attempt to write functions with this kind of behavior, we'll end up with a nested series of "if Nothing then Nothing else do something with the x in Just x" cases that will quickly become unwieldy:

add :: Maybe Int -> Maybe Int -> Maybe Int
add mx my =
  case mx of
    Nothing -> Nothing
    Just x  -> case my of
                 Nothing -> Nothing
                 Just y  -> Just (x + y)

To alleviate this, we can define operations for chaining these computations together. The bind binary operator (>>=) chains the results of one computation that could fail into a function that chooses another computation that could fail. If the first argument is Nothing, the second argument (the function) is ignored and the entire operation simply fails. If the first argument is Just x, we pass x to the function to get a new Maybe value, which may or may not result in a Just value.

(>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
Nothing  >>= _  =  Nothing
(Just x) >>= f  =  f x

We already have a value constructor that returns a value without affecting the computation's additional state: Just.

return :: a -> Maybe a
return x = Just x

We can then write the example as:

add :: Maybe Int -> Maybe Int -> Maybe Int
add mx my =
  mx >>= (\x ->
    my >>= (\y ->
      return (x + y)))

Using some additional syntactic sugar known as do-notation, the example can be written as:

add :: Maybe Int -> Maybe Int -> Maybe Int
add mx my = do
  x <- mx
  y <- my
  return (x + y)

Since this type of operation is quite common, there is a standard function in Haskell (liftM2) to take two monadic values (here: two Maybes) and combine their contents (two numbers) using another function (addition), making it possible to write the previous example as

add :: Maybe Int -> Maybe Int -> Maybe Int
add = liftM2 (+)

(Writing out the definition of liftM2 yields the code presented above in do-notation.)

The Writer monad

The Writer monad allows a process to carry additional information "on the side", along with the computed value. This can be useful to log error or debugging information which is not the primary result.

The following example implements a Writer monad in Javascript:

First, a Writer monad is declared. Function unit creates a new value type from a basic type, with an empty log string attached to it; and bind applies a function to an old value, and returns the new result value with an expanded log. The array brackets work here as the monad's type constructor, creating a value of the monadic type for the Writer monad from simpler components (the value in position 0 of the array, and the log string in position 1).

var unit = function(value) { return [value, ''] };

var bind = function(monadicValue, transformWithLog) {
    var value  = monadicValue[0],
        log = monadicValue[1],
        result = transformWithLog(value);
    return [ result[0], log + result[1] ];
};

pipeline is an auxiliary function that concatenates a sequence of binds applied to an array of functions.

var pipeline = function(monadicValue, functions) {
    for (var key in functions) {
        monadicValue = bind(monadicValue, functions[key]);
    }
    return monadicValue;
};

Examples of functions that return values of the type expected by the above Writer monad:

var squared = function(x) {
    return [x * x, 'was squared.'];
};

var halved = function(x) {
    return [x / 2, 'was halved.'];
};

Finally, an example of using the monad to build a pipeline of mathematical functions with debug information on the side:Template:Clarify

pipeline(unit(4), [squared, halved]); // [8, "was squared.was halved."]

The I/O monad

In a purely functional language, such as Haskell, functions cannot have any externally visible side effects as part of the function semantics. Although a function cannot directly cause a side effect, it can construct a value describing a desired side effect, that the caller should apply at a convenient time. In the Haskell notation, a value of type IO a represents an action that, when performed, produces a value of type a.

We can think of a value of type IO as a function that takes as its argument the current state of the world, and will return a new world where the state has been changed according to the function's return value. The state created in this way can be passed to another function, thus defining a series of functions which will be applied in order as steps of state changes. This process is similar to how a temporal logic represents the passage of time using only declarative propositions.

We would like to be able to describe all of the basic types of I/O operations, e.g. write text to standard output, read text from standard input, read and write files, send data over networks, etc. In addition, we need to be able to compose these primitives to form larger programs. For example, we would like to be able to write:

main :: IO ()
main = do
  putStrLn "What is your name?"
  name <- getLine
  putStrLn ("Nice to meet you, " ++ name ++ "!")

How can we formalize this intuitive notation? To do this, we need to be able to perform some basic operations with I/O actions:

• We should be able to sequence two I/O operations together. In Haskell, this is written as an infix operator >>, so that putStrLn "abc" >> putStrLn "def" is an I/O action that prints two lines of text to the console. The type of >> is IO a → IO b → IO b, meaning that the operator takes two I/O operations and returns a third that sequences the two together and returns the value of the second.
• We should have an I/O action which does nothing. That is, it returns a value but has no side effects. In Haskell, this action constructor is called return; it has type a → IO a.
• More subtly, we should be able to determine our next action based on the results of previous actions. To do this, Haskell has an operator >>= (pronounced bind) with type IO a → (a → IO b) → IO b. That is, the operand on the left is an I/O action that returns a value of type a; the operand on the right is a function that can pick an I/O action based on the value produced by the action on the left. The resulting combined action, when performed, performs the first action, then evaluates the function with the first action's return value, then performs the second action, and finally returns the second action's value.

An example of the use of this operator in Haskell would be getLine >>= putStrLn, which reads a single line of text from standard input and echos it to standard output. Note that the first operator, >>, is just a special case of this operator in which the return value of the first action is ignored and the selected second action is always the same.

It is not necessarily obvious that the three preceding operations, along with a suitable primitive set of I/O operations, allow us to define any program action whatsoever, including data transformations (using lambda expressions), if/then control flow, and looping control flows (using recursion). We can write the above example as one long expression:

main =
  putStrLn "What is your name?" >> 
  getLine >>= \name ->
  putStrLn ("Nice to meet you, " ++ name ++ "!")

The pipeline structure of the bind operator ensures that the getLine and putStrLn operations get evaluated only once and in the given order, so that the side-effects of extracting text from the input stream and writing to the output stream are correctly handled in the functional pipeline. This remains true even if the language performs out-of-order or lazy evaluation of functions.

Clearly, there is some common structure between the I/O definitions and the Maybe definitions, even though they are different in many ways. Monads are an abstraction upon the structures described above, and many similar structures, that finds and exploits the commonalities. The general monad concept includes any situation where the programmer wants to carry out a purely functional computation while a related computation is carried out on the side.

Formal definition

A monad is a construction that, given an underlying type system, embeds a corresponding type system (called the monadic type system) into it (that is, each monadic type acts as the underlying type). This monadic type system preserves all significant aspects of the underlying type system, while adding features particular to the monad.

The usual formulation of a monad for programming is known as a Kleisli triple, and has the following components:

1. A type constructor that defines, for every underlying type, how to obtain a corresponding monadic type. In Haskell's notation, the name of the monad represents the type constructor. If M is the name of the monad and t is a data type, then M t is the corresponding type in the monad.
2. A unit function that maps a value in an underlying type to a value in the corresponding monadic type. The unit function has the polymorphic type t→M t. The result is normally the "simplest" value in the corresponding type that completely preserves the original value (simplicity being understood appropriately to the monad). In Haskell, this function is called return due to the way it is used in the do-notation described later.
3. A binding operation of polymorphic type (M t)→(t→M u)→(M u), which Haskell represents by the infix operator >>=. Its first argument is a value in a monadic type, its second argument is a function that maps from the underlying type of the first argument to another monadic type, and its result is in that other monadic type. Typically, the binding operation can be understood as having four stages:
   1. The monad-related structure on the first argument is "pierced" to expose any number of values in the underlying type t.
   2. The given function is applied to all of those values to obtain values of type (M u).
   3. The monad-related structure on those values is also pierced, exposing values of type u.
   4. Finally, the monad-related structure is reassembled over all of the results, giving a single value of type (M u).

Given a type constructor M, in most contexts, a value of type M a can be thought of as an action that returns a value of type a. The return operation takes a value from a plain type a and puts it into a monadic container of type M a; the bind operation chains a monadic value of type M a with a function of type a → M b to create a monadic value of type M b.

Monad laws

For a monad to behave correctly, the definitions must obey a few axioms, together called the monad laws. The ≡ symbol indicates equivalence between two Haskell expressions in the following text.

• return acts approximately as a neutral element of >>=, in that:
  • (return x) >>= f ≡ f x
  • m >>= return ≡ m
• Binding two functions in succession is the same as binding one function that can be determined from them:
  • (m >>= f) >>= g ≡ m >>= ( \x -> (f x >>= g) )

The axioms can also be expressed using expressions in do-block style:

• do { f x } ≡ do { v <- return x; f v }
• do { m } ≡ do { v <- m; return v }
• do { x <- m; y <- f x; g y } ≡ do { y <- do { x <- m; f x }; g y }

or using the monadic composition operator, (f >=> g) x = (f x) >>= g:

• return >=> g ≡ g
• f >=> return ≡ f
• (f >=> g) >=> h ≡ f >=> (g >=> h)

fmap and join

Although Haskell defines monads in terms of the return and bind functions, it is also possible to define a monad in terms of return and two other operations, join and fmap. This formulation fits more closely with the definition of monads in category theory. The fmap operation, with type (t→u) → M t→M u, takes a function between two types and produces a function that does the "same thing" to values in the monad. The join operation, with type M (M t)→M t, "flattens" two layers of monadic information into one.

The two formulations are related as follows:

fmap f m = m >>= (return . f)
join n = n >>= id

m >>= g   ≡   join ((fmap g) m)

Here, m has the type M t, n has the type M (M r), f has the type t → u, and g has the type t → M v, where t, r, u and v are underlying types.

The fmap function is defined for any functor in the category of types and functions, not just for monads. It is expected to satisfy the functor laws:

fmap id   ≡   id
fmap (f . g)   ≡   (fmap f) . (fmap g)

The return function characterizes pointed functors in the same category, by accounting for the ability to "lift" values into the functor. It should satisfy the following law:

return . f   ≡   fmap f . return

In addition, the join function characterizes monads:

join . fmap join = join . join
join . fmap return = join . return = id
join . fmap (fmap f) = fmap f . join

Additive monads <Additive monad>...</Additive monad>

An additive monad is a monad endowed with a monadic zero mzero and a binary operator mplus satisfying the monoid laws, with the monadic zero as unit. The operator mplus has type M t → M t → M t (where M is the monad constructor and t is the underlying data type), satisfies the associative law and has the zero as both left and right identity. (Thus, an additive monad is also a monoid.)

Binding mzero with any function produces the zero for the result type, just as 0 multiplied by any number is 0.

mzero >>= f   ≡   mzero

Similarly, binding any m with a function that always returns a zero results in a zero

m >>= (\x -> mzero)   ≡   mzero

Intuitively, the zero represents a value in the monad that has only monad-related structure and no values from the underlying type. In the Maybe monad, "Nothing" is a zero. In the List monad, "[]" (the empty list) is a zero.

Syntactic sugar: do-notation <do-notation>...</do-notation>

Although there are times when it makes sense to use the bind operator >>= directly in a program, it is more typical to use a format called do-notation (perform-notation in OCaml, computation expressions in F#), that mimics the appearance of imperative languages. The compiler translates do-notation to expressions involving >>=. For example, the following code:

a = do x <- [3..4]
       [1..2]
       return (x, 42)

is transformed during compilation into:

a = [3..4] >>= (\x -> [1..2] >>= (\_ -> return (x, 42)))

It is helpful to see the implementation of the list monad, and to know that concatMap maps a function over a list and concatenates (flattens) the resulting lists:

instance Monad [] where
  m >>= f  = concatMap f m
  return x = [x]
  fail s   = []

Therefore, the following transformations hold and all the following expressions are equivalent:

a = [3..4] >>= (\x -> [1..2] >>= (\_ -> return (x, 42)))
a = [3..4] >>= (\x -> concatMap (\_ -> return (x, 42)) [1..2] )
a = [3..4] >>= (\x -> [(x,42),(x,42)] )
a = concatMap (\x -> [(x,42),(x,42)] ) [3..4]
a = [(3,42),(3,42),(4,42),(4,42)]

Notice that the list [1..2] is not used. The lack of a left-pointing arrow, translated into a binding to a function that ignores its argument, indicates that only the monadic structure is of interest, not the values inside it, e.g. for a state monad this might be used for changing the state without producing any more result values. The do-block notation can be used with any monad as it is simply syntactic sugar for >>=.

The following definitions for safe division for values in the Maybe monad are also equivalent:

x // y = do
  a <- x  -- Extract the values "inside" x and y, if there are any.
  b <- y
  if b == 0 then Nothing else Just (a / b)

x // y = x >>= (\a -> y >>= (\b -> if b == 0 then Nothing else Just (a / b)))

A similar example in F# using a computation expression:

let readNum () =
  let s = Console.ReadLine()
  let succ,v = Int32.TryParse(s)
  if (succ) then Some(v) else None

let secure_div = 
  maybe { 
    let! x = readNum()
    let! y = readNum()
    if (y = 0) 
    then None
    else return (x / y)
  }

The syntactic sugar of the maybe block would get translated internally to the following expression:

maybe.Delay(fun () ->
  maybe.Bind(readNum(), fun x ->
    maybe.Bind(readNum(), fun y ->
      if (y=0) then None else maybe.Return( x/y ))))

Generic monadic functions

Given values produced by safe division, we might want to carry on doing calculations without having to check manually if they are Nothing (i.e. resulted from an attempted division by zero). We can do this using a "lifting" function, which we can define not only for Maybe but for arbitrary monads. In Haskell this is called liftM2:

liftM2 :: Monad m => (a -> b -> c) -> m a -> m b -> m c
liftM2 op mx my = do
    x <- mx
    y <- my
    return (op x y)

Recall that arrows in a type associate to the right, so liftM2 is a function that takes a binary function as an argument and returns another binary function. The type signature says: If m is a monad, we can "lift" any binary function into it. For example:

(.*.) :: (Monad m, Num a) => m a -> m a -> m a
x .*. y = liftM2 (*) x y

defines an operator (.*.) which multiplies two numbers, unless one of them is Nothing (in which case it again returns Nothing). The advantage here is that we need not dive into the details of the implementation of the monad; if we need to do the same kind of thing with another function, or in another monad, using liftM2 makes it immediately clear what is meant (see Code reuse).

Mathematically, the liftM2 operator is defined by:

$\text{liftM2}:\mathrm{\forall }M:\text{monad},(A_1\to A_2\to R)\to M{A}_1\to M{A}_2\to MR=$$op\mapsto m_1\mapsto m_2\mapsto \text{bind}{m}_1(a_1\mapsto \text{bind}{m}_2(a_2\mapsto \text{return}(op{a}_1{a}_2)))$

Other examples

Identity monad

The simplest monad is the identity monad, which attaches no information to values.

Id t = t
return x = x
x >>= f = f x

A do-block in this monad performs variable substitution; do {x <- 2; return 3*x} results in 6.

From the category theory point of view, the identity monad is derived from the adjunction between identity functors.

Collections

Some familiar collection types, including lists, sets, and multisets, are monads. The definition for lists is given here.

-- "return" constructs a one-item list.
return x = [x]
-- "bind" concatenates the lists obtained by applying f to each item in list xs.
xs >>= f = concat (map f xs)
-- The zero object is an empty list.
mzero = []

List comprehensions are a special application of the list monad. For example, the list comprehension [ 2*x | x <- [1..n], isOkay x] corresponds to the computation in the list monad do {x <- [1..n]; if isOkay x then return (2*x) else mzero;}.

The notation of list comprehensions is similar to the set-builder notation, but sets can't be made into a monad, since there's a restriction on the type of computation to be comparable for equality, whereas a monad does not put any constraints on the types of computations. Actually, the Set is a restricted monad. The monads for collections naturally represent nondeterministic computation. The list (or other collection) represents all the possible results from different nondeterministic paths of computation at that given time. For example, when one executes x <- [1,2,3,4,5], one is saying that the variable x can non-deterministically take on any of the values of that list. If one were to return x, it would evaluate to a list of the results from each path of computation. Notice that the bind operator above follows this theme by performing f on each of the current possible results, and then it concatenates the result lists together.

Statements like if condition x y then return () else mzero are also often seen; if the condition is true, the non-deterministic choice is being performed from one dummy path of computation, which returns a value we are not assigning to anything; however, if the condition is false, then the mzero = [] monad value non-deterministically chooses from 0 values, effectively terminating that path of computation. Other paths of computations might still succeed. This effectively serves as a "guard" to enforce that only paths of computation that satisfy certain conditions can continue. So collection monads are very useful for solving logic puzzles, Sudoku, and similar problems.

In a language with lazy evaluation, like Haskell, a list is evaluated only to the degree that its elements are requested: for example, if one asks for the first element of a list, only the first element will be computed. With respect to usage of the list monad for non-deterministic computation that means that we can non-deterministically generate a lazy list of all results of the computation and ask for the first of them, and only as much work will be performed as is needed to get that first result. The process roughly corresponds to backtracking: a path of computation is chosen, and then if it fails at some point (if it evaluates mzero), then it backtracks to the last branching point, and follows the next path, and so on. If the second element is then requested, it again does just enough work to get the second solution, and so on. So the list monad is a simple way to implement a backtracking algorithm in a lazy language.

From the category theory point of view, collection monads are derived from adjunctions between a free functor and an underlying functor between the category of sets and a category of monoids. Taking different types of monoids, we obtain different types of collections.

type of collections	type of monoids
list	monoid
finite multiset	commutative monoid
finite set	idempotent commutative monoid
finite permutation	idempotent non-commutative monoid

State monads

A state monad allows a programmer to attach state information of any type to a calculation. Given any value type, the corresponding type in the state monad is a function which accepts a state, then outputs a new state (of type s) along with a return value (of type t).

type State s t = s -> (t, s)

Note that this monad, unlike those already seen, takes a type parameter, the type of the state information. The monad operations are defined as follows:

-- "return" produces the given value without changing the state.
return x = \s -> (x, s)
-- "bind" modifies m so that it applies f to its result.
m >>= f = \r -> let (x, s) = m r in (f x) s

Useful state operations include:

get = \s -> (s, s) -- Examine the state at this point in the computation.
put s = \_ -> ((), s) -- Replace the state.
modify f = \s -> ((), f s) -- Update the state

Another operation applies a state monad to a given initial state:

runState :: State s a -> s -> (a, s)
runState t s = t s

do-blocks in a state monad are sequences of operations that can examine and update the state data.

Informally, a state monad of state type S maps the type of return values T into functions of type $S\to T\times S$, where S is the underlying state. The return function is simply:

$\text{return}:T\to S\to T\times S=t\mapsto s\mapsto (t,s)$

The bind function is:

$\text{bind}:(S\to T\times S)\to (T\to S\to T^{\prime }\times S)\to S\to T^{\prime }\times S$$=m\mapsto k\mapsto s\mapsto (kt{s}^{\prime })\text{where}(t,s^{\prime })=ms$.

From the category theory point of view, a state monad is derived from the adjunction between the product functor and the exponential functor, which exists in any cartesian closed category by definition.

Environment monad

The environment monad (also called the reader monad and the function monad) allows a computation to depend on values from a shared environment. The monad type constructor maps a type T to functions of type E → T, where E is the type of the shared environment. The monad functions are:

$\text{return}:T\to E\to T=t\mapsto e\mapsto t$

$\text{bind}:(E\to T)\to (T\to E\to T^{\prime })\to E\to T^{\prime }=r\mapsto f\mapsto e\mapsto f(re)e$

The following monadic operations are useful:

$\text{ask}:E\to E={\text{id}}_E$

$\text{local}:(E\to E)\to (E\to T)\to E\to T=f\mapsto c\mapsto e\mapsto c(fe)$

The ask operation is used to retrieve the current context, while local executes a computation in a modified subcontext. As in the state monad, computations in the environment monad may be invoked by simply providing an environment value and applying it to an instance of the monad.

Writer monad

The writer monad allows a program to compute various kinds of auxiliary output which can be "composed" or "accumulated" step-by-step, in addition to the main result of a computation. It is often used for logging or profiling. Given the underlying type T, a value in the writer monad has type W × T, where W is a type endowed with an operation satisfying the monoid laws. The monad functions are simply:

$\text{return}:T\to W\times T=t\mapsto (ϵ,t)$

$\text{bind}:(W\times T)\to (T\to W\times T^{\prime })\to W\times T^{\prime }=(w,t)\mapsto f\mapsto (w*w^{\prime },t^{\prime })\text{where}(w^{\prime },t^{\prime })=ft$

where ε and * are the identity element of the monoid W and its associative operation, respectively.

The tell monadic operation is defined by:

$\text{tell}:W\to (W\times 1)=w\mapsto (w,())$

where 1 and () denote the unit type and its trivial element. It is used in combination with bind to update the auxiliary value without affecting the main computation.

Continuation monad

A continuation monad with return type $R$ maps type $T$ into functions of type $(T\to R)\to R$. It is used to model continuation-passing style. The return and bind functions are as follows:

$\text{return}:T\to (T\to R)\to R=t\mapsto f\mapsto ft$

$\text{bind}:((T\to R)\to R)\to (T\to (T^{\prime }\to R)\to R)\to (T^{\prime }\to R)\to R$$=c\mapsto f\mapsto k\mapsto c(t\mapsto ftk)$

The call-with-current-continuation function is defined as follows:

$\text{call/cc}:((T\to (T^{\prime }\to R)\to R)\to (T\to R)\to R)\to (T\to R)\to R$$=f\mapsto k\mapsto \left(f(t\mapsto x\mapsto kt)k\right)$

Others

Other concepts that researchers have expressed as monads include:

• Iteratee
• Exception handling
• Graphical user interfaces
• Interprocess communication
• Parsers
• Interpreters
• Strict evaluation
• Interfaces to code written in other languages

Comonads

Comonads are the categorical dual of monads. They are defined by a type constructor W T and two operations: extract with type W T → T for any T, and extend with type (W T → T' ) → W T → W T' . The operations extend and extract are expected to satisfy these laws:

$\text{extend}\text{extract}=\text{id}$

$\text{extract}\circ (\text{extend}f)=f$

$(\text{extend}f)\circ (\text{extend}g)=\text{extend}(f\circ (\text{extend}g))$

Alternatively, comonads may be defined in terms of operations fmap, extract and duplicate. The fmap and extract operations define W as a copointed functor. The duplicate operation characterizes comonads: it has type W T → W (W T) and satisfies the following laws:

$\text{extract}\circ \text{duplicate}=\text{id}$

$\text{fmap}\text{extract}\circ \text{duplicate}=\text{id}$

$\text{duplicate}\circ \text{duplicate}=\text{fmap}\text{duplicate}\circ \text{duplicate}$

The two formulations are related as follows:

$\text{fmap}:(A\to B)\to \mathrm{W}A\to \mathrm{W}B=f\mapsto \text{extend}(f\circ \text{extract})$

$\text{duplicate}:\mathrm{W}A\to \mathrm{W}\mathrm{W}A=\text{extend}\text{id}$

$\text{extend}:(\mathrm{W}A\to B)\to \mathrm{W}A\to \mathrm{W}B=f\mapsto (\text{fmap}f)\circ \text{duplicate}$

Whereas monads could be said to represent side-effects, a comonads W represents a kind of context. The extract functions extracts a value from its context, while the extend function may be used to compose a pipeline of "context-dependent functions" of type W A → B.

Identity comonad

The identity comonad is the simplest comonad: it maps type T to itself. The extract operator is the identity and the extend operator is function application.

Product comonad

The product comonad maps type $T$ into tuples of type $C\times T$, where $C$ is the context type of the comonad. The comonad operations are:

$\text{extract}:(C\times T)\to T=(c,t)\mapsto t$

$\text{extend}:((C\times A)\to B)\to C\times A\to C\times B=f\mapsto (c,a)\mapsto (c,f(c,a))$

$\text{fmap}:(A\to B)\to (C\times A)\to (C\times B)=(c,a)\mapsto (c,fa)$

$\text{duplicate}:(C\times A)\to (C\times (C\times A))=(c,a)\mapsto (c,(c,a))$

Function comonad

The function comonad maps type $T$ into functions of type $M\to T$, where $M$ is a type endowed with a monoid structure. The comonad operations are:

$\text{extract}:(M\to T)\to T=f\mapsto f\epsilon$

$\text{extend}:((M\to A)\to B)\to (M\to A)\to M\to B=f\mapsto g\mapsto m\mapsto f(m^{\prime }\mapsto g(m*m^{\prime }))$

$\text{fmap}:(A\to B)\to (M\to A)\to M\to B=f\mapsto g\mapsto (f\circ g)$

$\text{duplicate}:(M\to A)\to M\to (M\to A)=f\mapsto m\mapsto m^{\prime }\mapsto f(m*m^{\prime })$

where ε is the identity element of $M$ and * is its associative operation.

Costate comonad

The costate comonad maps a type $T$ into type $(S\to T)\times S$, where S is the base type of the store. The comonad operations are:

$\text{extract}:((S\to T)\times S)\to T=(f,s)\mapsto fs$

$\text{extend}:(((S\to A)\times S)\to B)\to ((S\to A)\times S)\to (S\to B)\times S$$=f\mapsto (g,s)\mapsto ((s^{\prime }\mapsto f(g,s^{\prime })),s)$

$\text{fmap}:(A\to B)\to ((S\to A)\times S)\to (S\to B)\times S=f\mapsto (f^{\prime },s)\mapsto (f\circ f^{\prime },s)$

$\text{duplicate}:((S\to A)\times S)\to (S\to ((S\to A)\times S))\times S=(f,s)\mapsto (s^{\prime }\mapsto (f,s^{\prime }),s)$

See also

• Arrows in functional programming — whereas monads generalize the results of a computation to effects, arrows further generalize the inputs similarly.
• Aspect-oriented programming, a paradigm to increase modularity by isolating secondary or supporting functionality.
• Effect system, an alternative way of describing side effects as types.
• Inversion of control — the abstract principle of calling specific functions from a reusable software entity.
• Monad transformers — which allow monads to be composed in a modular and convenient way.
• Uniqueness types - an alternative way of dealing with side-effects in functional languages

Notes

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External links

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Haskell monad tutorials

• Monad Tutorials Timeline Probably the most comprehensive collection of links to monad tutorials, ordered by date.
• Template:Cite web — The most famous "blog post" tutorial.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang — An attempt to explain all of the leading typeclasses in Haskell in an elementary way, with monadic functors considered as only one form, best understood by comparison with others: e.g., the more general idea of a "Functor" as something you can map over; "Applicative" functors, and so forth; contains an extensive bibliography.
• Template:Cite web — Opposes the idea of making a tutorial about monads in particular.
• What a Monad is not deals with common misconceptions and oversimplifications in a humorous way.
• Template:Cite web — Takes a similar point of view, locating monads in a much wider array of Haskell functor classes, of use only in special circumstances.
• Template:Cite web — An extremely detailed set of tutorials, deriving monads from first principles.
• Template:Cite web An explanation of Monads, building on the concepts of Functors, Applicative Functors and Monoids discussed in the previous chapter.
• Functors, Applicatives and Monads in Pictures. A humorous beginner's guide to monads.

Older tutorials

• All About Monads
• Haskell Wiki: Monads as Computation
• Haskell Wiki: Monads as Containers
• Template:Cite web
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Template:Cite web
• Monads A monad tutorial providing examples of non-trivial monads apart from the conventional IO/Maybe/List/State monads.
• Template:Cite web

Other documentation

• Template:Cite web
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang — The original paper suggesting use of monads for programming
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang — Describes monads in Haskell (before they were implemented)

Scala monad tutorials

• Template:Cite video
• Template:Cite web
• Template:Cite web
• Template:Cite web

Monads in other languages

• Monads in Perl
• Monads in Ruby
• Monads in Python
• Monads in Scala
• Monads in Clojure
• Monads in JavaScript
• Monads in C# and LINQ
• Monads in Ocaml
• Monads in PHP