-- Copyright : (c) Ross Paterson 2002
-- License : BSD-style (see the LICENSE file in the distribution)
--
--- Maintainer : ross@soi.city.ac.uk
+-- Maintainer : libraries@haskell.org
-- 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.
+-- /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.
+-- /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
-- <http://www.haskell.org/arrows/>.
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
+ -- * 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 <<<
+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
+--
+-- Minimal complete definition: 'arr' and 'first'.
+--
+-- 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
- f >>> g = g . f
- first f = f *** id
- second f = id *** f
- (f *** g) ~(x,y) = (f x, g y)
+ 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 (a -> m b)
+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)
- 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))
+ 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
--- | Right-to-left composition, for a better fit with arrow notation.
+-- | 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
-(<<<) :: Arrow a => a c d -> a b c -> a b d
-f <<< g = g >>> 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
+ zeroArrow :: a b c
instance MonadPlus m => ArrowZero (Kleisli m) where
- zeroArrow = Kleisli (\x -> mzero)
+ zeroArrow = Kleisli (\_ -> mzero)
class ArrowZero a => ArrowPlus a where
- (<+>) :: a b c -> a b c -> a b c
+ (<+>) :: 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)
+ 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.
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
+ -- | 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
+ 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)
+ 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
+ app :: a (a b c, b) c
instance ArrowApply (->) where
- app (f,x) = f x
+ app (f,x) = f x
instance Monad m => ArrowApply (Kleisli m) where
- app = Kleisli (\(Kleisli f, x) -> f x)
+ 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)
+newtype 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)
+ return x = ArrowMonad (arr (\_ -> 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
+ (\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
+ loop :: a (b,d) (c,d) -> a b c
instance ArrowLoop (->) where
- loop f b = let (c,d) = f (b,d) in c
+ 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)
+ loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
+ where f' x y = f (x, snd y)
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