Control.Monad.Free

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.

Instances of Monad should satisfy the following:

Left identity

return a >>= k = k a

Right identity

m >>= return = m

Associativity

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

Furthermore, the Monad and Applicative operations should relate as follows:

• pure = return

• m1 <*> m2 = m1 >>= (\x1 -> m2 >>= (\x2 -> return (x1 x2)))

The above laws imply:

• fmap f xs  =  xs >>= return . f

• (>>) = (*>)

and that pure and (<*>) satisfy the applicative functor laws.

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

A type f is a Functor if it provides a function fmap which, given any types a and b lets you apply any function from (a -> b) to turn an f a into an f b, preserving the structure of f. Furthermore f needs to adhere to the following:

Identity

fmap id == id

Composition

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

Note, that the second law follows from the free theorem of the type fmap and the first law, so you need only check that the former condition holds. See these articles by School of Haskell or David Luposchainsky for an explanation.

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

as >>= f == f =<< as

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.

'join bss' can be understood as the do expression

do bs <- bss
   bs

Examples

>>> join [[1, 2, 3], [4, 5, 6], [7, 8, 9]][1,2,3,4,5,6,7,8,9]

>>> join (Just (Just 3))Just 3

A common use of join is to run an IO computation returned from an GHC.Conc.STM transaction, since GHC.Conc.STM transactions can't perform IO directly. Recall that

GHC.Internal.Conc.atomically :: STM a -> IO a

is used to run GHC.Conc.STM transactions atomically. So, by specializing the types of GHC.Internal.Conc.atomically and join to

GHC.Internal.Conc.atomically :: STM (IO b) -> IO (IO b)
join       :: IO (IO b)  -> IO b

we can compose them as

join . GHC.Internal.Conc.atomically :: STM (IO b) -> IO b

to run an GHC.Conc.STM transaction and the IO action it returns.

Promote a function to a monad. This is equivalent to fmap but specialised to Monads.

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

liftM<n> f x1 x2 ... xn

Examples

>>> pure (\x y z -> x + y * z) `ap` Just 1 `ap` Just 5 `ap` Just 10Just 51

Monads that also support choice and failure.

Map each element of a structure to a monadic action, evaluate these actions from left to right, and ignore the results. For a version that doesn't ignore the results see Data.Traversable.mapM.

mapM_ is just like traverse_, but specialised to monadic actions.

Evaluate each monadic action in the structure from left to right, and ignore the results. For a version that doesn't ignore the results see Data.Traversable.sequence.

sequence_ is just like sequenceA_, but specialised to monadic actions.

Map each element of a structure to a monadic action, evaluate these actions from left to right, and collect the results. For a version that ignores the results see Data.Foldable.mapM_.

Examples

mapM is literally a traverse with a type signature restricted to Monad. Its implementation may be more efficient due to additional power of Monad.

Evaluate each monadic action in the structure from left to right, and collect the results. For a version that ignores the results see Data.Foldable.sequence_.

Examples

Basic usage:

The first two examples are instances where the input and and output of sequence are isomorphic.

>>> sequence $ Right [1,2,3,4][Right 1,Right 2,Right 3,Right 4]

>>> sequence $ [Right 1,Right 2,Right 3,Right 4]Right [1,2,3,4]

The following examples demonstrate short circuit behavior for sequence.

>>> sequence $ Left [1,2,3,4]Left [1,2,3,4]

>>> sequence $ [Left 0, Right 1,Right 2,Right 3,Right 4]Left 0

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

Examples

>>> liftM2 (+) [0,1] [0,2][0,2,1,3]

>>> liftM2 (+) (Just 1) NothingNothing

>>> liftM2 (+) (+ 3) (* 2) 518

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

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

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

Conditional execution of Applicative expressions. For example,

Examples

when debug (putStrLn "Debugging")

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

>>> putStr "pi:" >> when False (print 3.14159)pi:

Strict version of Data.Functor.<$>.

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

Note how this operator resembles function composition (.):

(.)   ::            (b ->   c) -> (a ->   b) -> a ->   c
(<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c

Left-to-right composition of Kleisli arrows.

'(bs >=> cs) a' can be understood as the do expression

do b <- bs a
   cs b

or in terms of (>>=) as

bs a >>= cs

This generalizes the list-based filter function.

runIdentity (filterM (Identity . p) xs) == filter p xs

Examples

>>> filterM (\x -> do      putStrLn ("Keep: " ++ show x ++ "?")      answer <- getLine      pure (answer == "y"))    [1, 2, 3]Keep: 1?yKeep: 2?nKeep: 3?y[1,3]

>>> filterM (\x -> do      putStr (show x)      x' <- readLn      pure (x == x'))    [1, 2, 3]122233[2,3]

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.

Note: foldM is the same as foldlM

Like foldM, but discards the result.

Repeat an action indefinitely.

Examples

A common use of forever is to process input from network sockets, System.IO.Handles, and channels (e.g. Control.Concurrent.MVar.MVar and Chan).

For example, here is how we might implement an echo server, using forever both to listen for client connections on a network socket and to echo client input on client connection handles:

echoServer :: Socket -> IO ()
echoServer socket = forever $ do
  client <- accept socket
  forkFinally (echo client) (\_ -> hClose client)
  where
    echo :: Handle -> IO ()
    echo client = forever $
      hGetLine client >>= hPutStrLn client

Note that "forever" isn't necessarily non-terminating. If the action is in a MonadPlus and short-circuits after some number of iterations. then forever actually returns mzero, effectively short-circuiting its caller.

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 monad.

Direct MonadPlus equivalent of filter.

Examples

The filter function is just mfilter specialized to the list monad:

filter = ( mfilter :: (a -> Bool) -> [a] -> [a] )

An example using mfilter with the Maybe monad:

>>> mfilter odd (Just 1)Just 1>>> mfilter odd (Just 2)Nothing

replicateM n act performs the action act n times, and then returns the list of results.

replicateM n (pure x) == replicate n x

Examples

>>> replicateM 3 getLinehiheyahiya["hi","heya","hiya"]

>>> import Control.Monad.State>>> runState (replicateM 3 $ state $ \s -> (s, s + 1)) 1([1,2,3],4)

Like replicateM, but discards the result.

Examples

>>> replicateM_ 3 (putStr "a")aaa

The reverse of when.

Examples

>>> do x <- getLine       unless (x == "hi") (putStrLn "hi!")comingupwithexamplesisdifficulthi!

>>> unless (pi > exp 1) NothingJust ()

The zipWithM function generalizes zipWith to arbitrary applicative functors.

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

forM_ is mapM_ with its arguments flipped. For a version that doesn't ignore the results see Data.Traversable.forM.

forM_ is just like for_, but specialised to monadic actions.

The sum of a collection of actions using (<|>), generalizing concat.

msum is just like asum, but specialised to MonadPlus.

Examples

Basic usage, using the MonadPlus instance for Maybe:

>>> msum [Just "Hello", Nothing, Just "World"]Just "Hello"

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

Examples

Replace the contents of a Maybe Int with unit:

>>> void NothingNothing

>>> void (Just 3)Just ()

Replace the contents of an Either Int Int with unit, resulting in an Either Int ():

>>> void (Left 8675309)Left 8675309

>>> void (Right 8675309)Right ()

Replace every element of a list with unit:

>>> void [1,2,3][(),(),()]

Replace the second element of a pair with unit:

>>> void (1,2)(1,())

Discard the result of an System.IO.IO action:

>>> mapM print [1,2]12[(),()]

>>> void $ mapM print [1,2]12

forM is mapM with its arguments flipped. For a version that ignores the results see Data.Foldable.forM_.

Conditional failure of Alternative computations. Defined by

guard True  = pure ()
guard False = empty

Examples

Common uses of guard include conditionally signalling an error in an error monad and conditionally rejecting the current choice in an Alternative-based parser.

As an example of signalling an error in the error monad Maybe, consider a safe division function safeDiv x y that returns Nothing when the denominator y is zero and Just (x `div` y) otherwise. For example:

>>> safeDiv 4 0Nothing

>>> safeDiv 4 2Just 2

A definition of safeDiv using guards, but not guard:

safeDiv :: Int -> Int -> Maybe Int
safeDiv x y | y /= 0    = Just (x `div` y)
            | otherwise = Nothing

A definition of safeDiv using guard and Monad do-notation:

safeDiv :: Int -> Int -> Maybe Int
safeDiv x y = do
  guard (y /= 0)
  return (x `div` y)

When a value is bound in do-notation, the pattern on the left hand side of <- might not match. In this case, this class provides a function to recover.

A Monad without a MonadFail instance may only be used in conjunction with pattern that always match, such as newtypes, tuples, data types with only a single data constructor, and irrefutable patterns (~pat).

Instances of MonadFail should satisfy the following law: fail s should be a left zero for >>=,

fail s >>= f  =  fail s

If your Monad is also MonadPlus, a popular definition is

fail _ = mzero

fail s should be an action that runs in the monad itself, not an exception (except in instances of MonadIO). In particular, fail should not be implemented in terms of error.

When a value is bound in do-notation, the pattern on the left hand side of <- might not match. In this case, this class provides a function to recover.

A Monad without a MonadFail instance may only be used in conjunction with pattern that always match, such as newtypes, tuples, data types with only a single data constructor, and irrefutable patterns (~pat).

Instances of MonadFail should satisfy the following law: fail s should be a left zero for >>=,

fail s >>= f  =  fail s

If your Monad is also MonadPlus, a popular definition is

fail _ = mzero

fail s should be an action that runs in the monad itself, not an exception (except in instances of MonadIO). In particular, fail should not be implemented in terms of error.

This type class generalizes over encodings of Free Monads.