Control/Arrow.hs

    {-# 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)