{-# LANGUAGE Trustworthy #-} ----------------------------------------------------------------------------- -- | -- Module : Control.Arrow -- Copyright : (c) Ross Paterson 2002 -- License : BSD-style (see the LICENSE file in the distribution) -- -- Maintainer : libraries@haskell.org -- Stability : provisional -- 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. -- -- These papers and more information on arrows can be found at -- <http://www.haskell.org/arrows/>. module Control.Arrow ( -- * Arrows Arrow(..), Kleisli(..), -- ** Derived combinators returnA, (^>>), (>>^), (>>>), (<<<), -- reexported -- ** Right-to-left variants (<<^), (^<<), -- * Monoid operations ArrowZero(..), ArrowPlus(..), -- * Conditionals ArrowChoice(..), -- * Arrow application ArrowApply(..), ArrowMonad(..), leftApp, -- * Feedback ArrowLoop(..) ) where import Prelude hiding (id,(.)) import Control.Monad import Control.Monad.Fix import Control.Category infixr 5 <+> infixr 3 *** infixr 3 &&& infixr 2 +++ infixr 2 ||| infixr 1 ^>>, >>^ infixr 1 ^<<, <<^ -- | The basic arrow class. -- -- Minimal complete definition: 'arr' and 'first', satisfying the laws -- -- * @'arr' id = 'id'@ -- -- * @'arr' (f >>> g) = 'arr' f >>> 'arr' g@ -- -- * @'first' ('arr' f) = 'arr' ('first' f)@ -- -- * @'first' (f >>> g) = 'first' f >>> 'first' g@ -- -- * @'first' f >>> 'arr' 'fst' = 'arr' 'fst' >>> f@ -- -- * @'first' f >>> 'arr' ('id' *** g) = 'arr' ('id' *** g) >>> 'first' f@ -- -- * @'first' ('first' f) >>> 'arr' 'assoc' = 'arr' 'assoc' >>> 'first' f@ -- -- where -- -- > assoc ((a,b),c) = (a,(b,c)) -- -- The other combinators have sensible default definitions, -- which may be overridden for efficiency. class Category a => Arrow a where -- | Lift a function to an arrow. arr :: (b -> c) -> a b c -- | 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) 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 {-# RULES "compose/arr" forall f g . (arr f) . (arr g) = arr (f . g) "first/arr" forall f . first (arr f) = arr (first f) "second/arr" forall f . second (arr f) = arr (second f) "product/arr" forall f g . arr f *** arr g = arr (f *** g) "fanout/arr" forall f g . arr f &&& arr g = arr (f &&& g) "compose/first" forall f g . (first f) . (first g) = first (f . g) "compose/second" forall f g . (second f) . (second g) = second (f . g) #-} -- Ordinary functions are arrows. instance Arrow (->) where arr f = 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 { runKleisli :: a -> m b } instance Monad m => Category (Kleisli m) where id = Kleisli return (Kleisli f) . (Kleisli g) = Kleisli (\b -> g b >>= f) instance Monad m => Arrow (Kleisli m) where arr f = Kleisli (return . f) 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 -- | Precomposition with a pure function. (^>>) :: Arrow a => (b -> c) -> a c d -> a b d f ^>> a = arr f >>> a -- | Postcomposition with a pure function. (>>^) :: Arrow a => a b c -> (c -> d) -> a b d a >>^ f = a >>> arr f -- | Precomposition with a pure function (right-to-left variant). (<<^) :: Arrow a => a c d -> (b -> c) -> a b d a <<^ f = a <<< arr f -- | Postcomposition with a pure function (right-to-left variant). (^<<) :: Arrow a => (c -> d) -> a b c -> a b d f ^<< a = arr f <<< a class Arrow a => ArrowZero a where zeroArrow :: a b c instance MonadPlus m => ArrowZero (Kleisli m) where zeroArrow = Kleisli (\_ -> mzero) -- | A monoid on arrows. class ArrowZero a => ArrowPlus a where -- | An associative operation with identity 'zeroArrow'. (<+>) :: 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. -- -- Minimal complete definition: 'left', satisfying the laws -- -- * @'left' ('arr' f) = 'arr' ('left' f)@ -- -- * @'left' (f >>> g) = 'left' f >>> 'left' g@ -- -- * @f >>> 'arr' 'Left' = 'arr' 'Left' >>> 'left' f@ -- -- * @'left' f >>> 'arr' ('id' +++ g) = 'arr' ('id' +++ g) >>> 'left' f@ -- -- * @'left' ('left' f) >>> 'arr' 'assocsum' = 'arr' 'assocsum' >>> 'left' f@ -- -- where -- -- > assocsum (Left (Left x)) = Left x -- > assocsum (Left (Right y)) = Right (Left y) -- > assocsum (Right z) = Right (Right z) -- -- 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 :: Either x y -> Either y x 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 {-# RULES "left/arr" forall f . left (arr f) = arr (left f) "right/arr" forall f . right (arr f) = arr (right f) "sum/arr" forall f g . arr f +++ arr g = arr (f +++ g) "fanin/arr" forall f g . arr f ||| arr g = arr (f ||| g) "compose/left" forall f g . left f . left g = left (f . g) "compose/right" forall f g . right f . right g = right (f . g) #-} 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. -- Instances should satisfy the following laws: -- -- * @'first' ('arr' (\\x -> 'arr' (\\y -> (x,y)))) >>> 'app' = 'id'@ -- -- * @'first' ('arr' (g >>>)) >>> 'app' = 'second' g >>> 'app'@ -- -- * @'first' ('arr' (>>> h)) >>> 'app' = 'app' >>> h@ -- -- Such arrows are equivalent to monads (see 'ArrowMonad'). 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 ArrowMonad a b = ArrowMonad (a () b) instance Arrow a => Functor (ArrowMonad a) where fmap f (ArrowMonad m) = ArrowMonad $ m >>> arr f instance ArrowApply a => Monad (ArrowMonad a) where return x = ArrowMonad (arr (\_ -> x)) ArrowMonad m >>= f = ArrowMonad $ m >>> arr (\x -> let ArrowMonad h = f x in (h, ())) >>> app instance (ArrowApply a, ArrowPlus a) => MonadPlus (ArrowMonad a) where mzero = ArrowMonad zeroArrow ArrowMonad x `mplus` ArrowMonad y = ArrowMonad (x <+> y) -- | 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, although the computation occurs only once. -- It underlies the @rec@ value recursion construct in arrow notation. -- 'loop' should satisfy the following laws: -- -- [/extension/] -- @'loop' ('arr' f) = 'arr' (\\ b -> 'fst' ('fix' (\\ (c,d) -> f (b,d))))@ -- -- [/left tightening/] -- @'loop' ('first' h >>> f) = h >>> 'loop' f@ -- -- [/right tightening/] -- @'loop' (f >>> 'first' h) = 'loop' f >>> h@ -- -- [/sliding/] -- @'loop' (f >>> 'arr' ('id' *** k)) = 'loop' ('arr' ('id' *** k) >>> f)@ -- -- [/vanishing/] -- @'loop' ('loop' f) = 'loop' ('arr' unassoc >>> f >>> 'arr' assoc)@ -- -- [/superposing/] -- @'second' ('loop' f) = 'loop' ('arr' assoc >>> 'second' f >>> 'arr' unassoc)@ -- -- where -- -- > assoc ((a,b),c) = (a,(b,c)) -- > unassoc (a,(b,c)) = ((a,b),c) -- 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 -- | Beware that for many monads (those for which the '>>=' operation -- is strict) this instance will /not/ satisfy the right-tightening law -- required by the 'ArrowLoop' class. instance MonadFix m => ArrowLoop (Kleisli m) where loop (Kleisli f) = Kleisli (liftM fst . mfix . f') where f' x y = f (x, snd y)