Control.Monad

The Functor, Monad and MonadPlus classes, with some useful operations on monads.

Functor and monad classes

class Functor f where

The Functor class is used for types that can be mapped over. Instances of Functor should satisfy the following laws:

fmap id  ==  id
fmap (f . g)  ==  fmap f . fmap g

The instances of Functor for lists, Maybe and IO satisfy these laws.

Methods

fmap ∷ (a → b) → f a → f b

Instances

Functor []

Functor IO

Functor Maybe

Functor ReadP

Functor ReadPrec

Functor STM

Functor Handler

Functor ZipList

Functor ArgDescr

Functor OptDescr

Functor ArgOrder

Functor ((→) r)

Functor (Either a)

Functor ((,) a)

Functor (ST s)

Functor (Proxy ★)

Arrow a ⇒ Functor (ArrowMonad a)

Functor (ST s)

Monad m ⇒ Functor (WrappedMonad m)

Functor (Const m)

Arrow a ⇒ Functor (WrappedArrow a b)

class Monad m where

The Monad class defines the basic operations over a monad, a concept from a branch of mathematics known as category theory. From the perspective of a Haskell programmer, however, it is best to think of a monad as an abstract datatype of actions. Haskell's do expressions provide a convenient syntax for writing monadic expressions.

Minimal complete definition: >>= and return.

Instances of Monad should satisfy the following laws:

return a >>= k  ==  k a
m >>= return  ==  m
m >>= (\x -> k x >>= h)  ==  (m >>= k) >>= h

Instances of both Monad and Functor should additionally satisfy the law:

fmap f xs  ==  xs >>= return . f

The instances of Monad for lists, Maybe and IO defined in the Prelude satisfy these laws.

Minimal complete definition

(>>=), return

Methods

(>>=) ∷ ∀ a b. m a → (a → m b) → m b infixl 1

Sequentially compose two actions, passing any value produced by the first as an argument to the second.

(>>) ∷ ∀ a b. m a → m b → m b infixl 1

Sequentially compose two actions, discarding any value produced by the first, like sequencing operators (such as the semicolon) in imperative languages.

return ∷ a → m a

Inject a value into the monadic type.

fail ∷ String → m a

Fail with a message. This operation is not part of the mathematical definition of a monad, but is invoked on pattern-match failure in a do expression.

Instances

Monad []

Monad IO

Monad Maybe

Monad ReadP

Monad ReadPrec

Monad STM

Monad ((→) r)

Monad (Either e)

Monad (ST s)

Monad (Proxy ★)

ArrowApply a ⇒ Monad (ArrowMonad a)

Monad (ST s)

Monad m ⇒ Monad (WrappedMonad m)

class Monad m ⇒ MonadPlus m where

Monads that also support choice and failure.

Methods

mzero ∷ m a

the identity of mplus. It should also satisfy the equations

mzero >>= f  =  mzero
v >> mzero   =  mzero

mplus ∷ m a → m a → m a

an associative operation

Instances

MonadPlus []

MonadPlus Maybe

MonadPlus ReadP

MonadPlus ReadPrec

MonadPlus STM

(ArrowApply a, ArrowPlus a) ⇒ MonadPlus (ArrowMonad a)

Functions

Naming conventions

The functions in this library use the following naming conventions:

• A postfix 'M' always stands for a function in the Kleisli category: The monad type constructor m is added to function results (modulo currying) and nowhere else. So, for example,

 filter  ::              (a ->   Bool) -> [a] ->   [a]
 filterM :: (Monad m) => (a -> m Bool) -> [a] -> m [a]

• A postfix '_' changes the result type from (m a) to (m ()). Thus, for example:

 sequence  :: Monad m => [m a] -> m [a] 
 sequence_ :: Monad m => [m a] -> m () 

• A prefix 'm' generalizes an existing function to a monadic form. Thus, for example:

 sum  :: Num a       => [a]   -> a
 msum :: MonadPlus m => [m a] -> m a

Basic Monad functions

mapM ∷ Monad m ⇒ (a → m b) → [a] → m [b]

mapM f is equivalent to sequence . map f.

mapM_ ∷ Monad m ⇒ (a → m b) → [a] → m ()

mapM_ f is equivalent to sequence_ . map f.

forM ∷ Monad m ⇒ [a] → (a → m b) → m [b]

forM is mapM with its arguments flipped

forM_ ∷ Monad m ⇒ [a] → (a → m b) → m ()

forM_ is mapM_ with its arguments flipped

sequence ∷ Monad m ⇒ [m a] → m [a]

Evaluate each action in the sequence from left to right, and collect the results.

sequence_ ∷ Monad m ⇒ [m a] → m ()

Evaluate each action in the sequence from left to right, and ignore the results.

(=<<) ∷ Monad m ⇒ (a → m b) → m a → m b infixr 1

Same as >>=, but with the arguments interchanged.

(>=>) ∷ Monad m ⇒ (a → m b) → (b → m c) → a → m c infixr 1

Left-to-right Kleisli composition of monads.

(<=<) ∷ Monad m ⇒ (b → m c) → (a → m b) → a → m c infixr 1

Right-to-left Kleisli composition of monads. (>=>), with the arguments flipped

forever ∷ Monad m ⇒ m a → m b

forever act repeats the action infinitely.

void ∷ Functor f ⇒ f a → f ()

void value discards or ignores the result of evaluation, such as the return value of an IO action.

Generalisations of list functions

join ∷ Monad m ⇒ m (m a) → m a

The join function is the conventional monad join operator. It is used to remove one level of monadic structure, projecting its bound argument into the outer level.

msum ∷ MonadPlus m ⇒ [m a] → m a

This generalizes the list-based concat function.

mfilter ∷ MonadPlus m ⇒ (a → Bool) → m a → m a

Direct MonadPlus equivalent of filter filter = (mfilter:: (a -> Bool) -> [a] -> [a] applicable to any MonadPlus, for example mfilter odd (Just 1) == Just 1 mfilter odd (Just 2) == Nothing

filterM ∷ Monad m ⇒ (a → m Bool) → [a] → m [a]

This generalizes the list-based filter function.

mapAndUnzipM ∷ Monad m ⇒ (a → m (b, c)) → [a] → m ([b], [c])

The mapAndUnzipM function maps its first argument over a list, returning the result as a pair of lists. This function is mainly used with complicated data structures or a state-transforming monad.

zipWithM ∷ Monad m ⇒ (a → b → m c) → [a] → [b] → m [c]

The zipWithM function generalizes zipWith to arbitrary monads.

zipWithM_ ∷ Monad m ⇒ (a → b → m c) → [a] → [b] → m ()

zipWithM_ is the extension of zipWithM which ignores the final result.

foldM ∷ Monad m ⇒ (a → b → m a) → a → [b] → m a

The foldM function is analogous to foldl, except that its result is encapsulated in a monad. Note that foldM works from left-to-right over the list arguments. This could be an issue where (>>) and the `folded function' are not commutative.

      foldM f a1 [x1, x2, ..., xm]

==

      do
        a2 <- f a1 x1
        a3 <- f a2 x2
        ...
        f am xm

If right-to-left evaluation is required, the input list should be reversed.

foldM_ ∷ Monad m ⇒ (a → b → m a) → a → [b] → m ()

Like foldM, but discards the result.

replicateM ∷ Monad m ⇒ Int → m a → m [a]

replicateM n act performs the action n times, gathering the results.

replicateM_ ∷ Monad m ⇒ Int → m a → m ()

Like replicateM, but discards the result.

Conditional execution of monadic expressions

guard ∷ MonadPlus m ⇒ Bool → m ()

guard b is return () if b is True, and mzero if b is False.

when ∷ Monad m ⇒ Bool → m () → m ()

Conditional execution of monadic expressions. For example,

      when debug (putStr "Debugging\n")

will output the string Debugging\n if the Boolean value debug is True, and otherwise do nothing.

unless ∷ Monad m ⇒ Bool → m () → m ()

The reverse of when.

Monadic lifting operators

liftM ∷ Monad m ⇒ (a1 → r) → m a1 → m r

Promote a function to a monad.

liftM2 ∷ Monad m ⇒ (a1 → a2 → r) → m a1 → m a2 → m r

Promote a function to a monad, scanning the monadic arguments from left to right. For example,

   liftM2 (+) [0,1] [0,2] = [0,2,1,3]
   liftM2 (+) (Just 1) Nothing = Nothing

liftM3 ∷ Monad m ⇒ (a1 → a2 → a3 → r) → m a1 → m a2 → m a3 → m r

Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

liftM4 ∷ Monad m ⇒ (a1 → a2 → a3 → a4 → r) → m a1 → m a2 → m a3 → m a4 → m r

Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

liftM5 ∷ Monad m ⇒ (a1 → a2 → a3 → a4 → a5 → r) → m a1 → m a2 → m a3 → m a4 → m a5 → m r

Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

ap ∷ Monad m ⇒ m (a → b) → m a → m b

In many situations, the liftM operations can be replaced by uses of ap, which promotes function application.

      return f `ap` x1 `ap` ... `ap` xn

is equivalent to

      liftMn f x1 x2 ... xn