-----------------------------------------------------------------------------
-- |
-- Module : Control.Arrow
-- Copyright : (c) Ross Paterson 2002
-- License : BSD-style (see the LICENSE file in the distribution)
--
-- Maintainer : ross@soi.city.ac.uk
-- Stability : experimental
-- Portability : portable
--
-- Basic arrow definitions, based on
-- /Generalising Monads to Arrows/, by John Hughes,
-- /Science of Computer Programming/ 37, pp67-111, May 2000.
-- plus a couple of definitions ('returnA' and 'loop') from
-- /A New Notation for Arrows/, by Ross Paterson, in /ICFP 2001/,
-- Firenze, Italy, pp229-240.
-- See these papers for the equations these combinators are expected to
-- satisfy. These papers and more information on arrows can be found at
-- .
module Control.Arrow (
-- * Arrows
Arrow(..), Kleisli(..),
-- ** Derived combinators
returnA, (<<<),
-- * Monoid operations
ArrowZero(..), ArrowPlus(..),
-- * Conditionals
ArrowChoice(..),
-- * Arrow application
ArrowApply(..), ArrowMonad(..), leftApp,
-- * Feedback
ArrowLoop(..)
) where
import Prelude
import Control.Monad
import Control.Monad.Fix
infixr 5 <+>
infixr 3 ***
infixr 3 &&&
infixr 2 +++
infixr 2 |||
infixr 1 >>>
infixr 1 <<<
-- | The basic arrow class.
-- Any instance must define either 'arr' or 'pure' (which are synonyms),
-- as well as '>>>' and 'first'. The other combinators have sensible
-- default definitions, which may be overridden for efficiency.
class Arrow a where
-- | Lift a function to an arrow: you must define either this
-- or 'pure'.
arr :: (b -> c) -> a b c
arr = pure
-- | A synonym for 'arr': you must define one or other of them.
pure :: (b -> c) -> a b c
pure = arr
-- | Left-to-right composition of arrows.
(>>>) :: a b c -> a c d -> a b d
-- | Send the first component of the input through the argument
-- arrow, and copy the rest unchanged to the output.
first :: a b c -> a (b,d) (c,d)
-- | A mirror image of 'first'.
--
-- The default definition may be overridden with a more efficient
-- version if desired.
second :: a b c -> a (d,b) (d,c)
second f = arr swap >>> first f >>> arr swap
where swap ~(x,y) = (y,x)
-- | Split the input between the two argument arrows and combine
-- their output. Note that this is in general not a functor.
--
-- The default definition may be overridden with a more efficient
-- version if desired.
(***) :: a b c -> a b' c' -> a (b,b') (c,c')
f *** g = first f >>> second g
-- | Fanout: send the input to both argument arrows and combine
-- their output.
--
-- The default definition may be overridden with a more efficient
-- version if desired.
(&&&) :: a b c -> a b c' -> a b (c,c')
f &&& g = arr (\b -> (b,b)) >>> f *** g
-- Ordinary functions are arrows.
instance Arrow (->) where
arr f = f
f >>> g = g . f
first f = f *** id
second f = id *** f
-- (f *** g) ~(x,y) = (f x, g y)
-- sorry, although the above defn is fully H'98, nhc98 can't parse it.
(***) f g ~(x,y) = (f x, g y)
-- | Kleisli arrows of a monad.
newtype Kleisli m a b = Kleisli (a -> m b)
instance Monad m => Arrow (Kleisli m) where
arr f = Kleisli (return . f)
Kleisli f >>> Kleisli g = Kleisli (\b -> f b >>= g)
first (Kleisli f) = Kleisli (\ ~(b,d) -> f b >>= \c -> return (c,d))
second (Kleisli f) = Kleisli (\ ~(d,b) -> f b >>= \c -> return (d,c))
-- | The identity arrow, which plays the role of 'return' in arrow notation.
returnA :: Arrow a => a b b
returnA = arr id
-- | Right-to-left composition, for a better fit with arrow notation.
(<<<) :: Arrow a => a c d -> a b c -> a b d
f <<< g = g >>> f
class Arrow a => ArrowZero a where
zeroArrow :: a b c
instance MonadPlus m => ArrowZero (Kleisli m) where
zeroArrow = Kleisli (\x -> mzero)
class ArrowZero a => ArrowPlus a where
(<+>) :: a b c -> a b c -> a b c
instance MonadPlus m => ArrowPlus (Kleisli m) where
Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
-- | Choice, for arrows that support it. This class underlies the
-- @if@ and @case@ constructs in arrow notation.
-- Any instance must define 'left'. The other combinators have sensible
-- default definitions, which may be overridden for efficiency.
class Arrow a => ArrowChoice a where
-- | Feed marked inputs through the argument arrow, passing the
-- rest through unchanged to the output.
left :: a b c -> a (Either b d) (Either c d)
-- | A mirror image of 'left'.
--
-- The default definition may be overridden with a more efficient
-- version if desired.
right :: a b c -> a (Either d b) (Either d c)
right f = arr mirror >>> left f >>> arr mirror
where mirror (Left x) = Right x
mirror (Right y) = Left y
-- | Split the input between the two argument arrows, retagging
-- and merging their outputs.
-- Note that this is in general not a functor.
--
-- The default definition may be overridden with a more efficient
-- version if desired.
(+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
f +++ g = left f >>> right g
-- | Fanin: Split the input between the two argument arrows and
-- merge their outputs.
--
-- The default definition may be overridden with a more efficient
-- version if desired.
(|||) :: a b d -> a c d -> a (Either b c) d
f ||| g = f +++ g >>> arr untag
where untag (Left x) = x
untag (Right y) = y
instance ArrowChoice (->) where
left f = f +++ id
right f = id +++ f
f +++ g = (Left . f) ||| (Right . g)
(|||) = either
instance Monad m => ArrowChoice (Kleisli m) where
left f = f +++ arr id
right f = arr id +++ f
f +++ g = (f >>> arr Left) ||| (g >>> arr Right)
Kleisli f ||| Kleisli g = Kleisli (either f g)
-- | Some arrows allow application of arrow inputs to other inputs.
class Arrow a => ArrowApply a where
app :: a (a b c, b) c
instance ArrowApply (->) where
app (f,x) = f x
instance Monad m => ArrowApply (Kleisli m) where
app = Kleisli (\(Kleisli f, x) -> f x)
-- | The 'ArrowApply' class is equivalent to 'Monad': any monad gives rise
-- to a 'Kleisli' arrow, and any instance of 'ArrowApply' defines a monad.
newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)
instance ArrowApply a => Monad (ArrowMonad a) where
return x = ArrowMonad (arr (\z -> x))
ArrowMonad m >>= f = ArrowMonad (m >>>
arr (\x -> let ArrowMonad h = f x in (h, ())) >>>
app)
-- | Any instance of 'ArrowApply' can be made into an instance of
-- 'ArrowChoice' by defining 'left' = 'leftApp'.
leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d)
leftApp f = arr ((\b -> (arr (\() -> b) >>> f >>> arr Left, ())) |||
(\d -> (arr (\() -> d) >>> arr Right, ()))) >>> app
-- | The 'loop' operator expresses computations in which an output value is
-- fed back as input, even though the computation occurs only once.
-- It underlies the @rec@ value recursion construct in arrow notation.
class Arrow a => ArrowLoop a where
loop :: a (b,d) (c,d) -> a b c
instance ArrowLoop (->) where
loop f b = let (c,d) = f (b,d) in c
instance MonadFix m => ArrowLoop (Kleisli m) where
loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
where f' x y = f (x, snd y)