-----------------------------------------------------------------------------
---
+-- |
-- Module : Control.Arrow
-- Copyright : (c) Ross Paterson 2002
-- License : BSD-style (see the LICENSE file in the distribution)
-- Stability : experimental
-- Portability : portable
--
--- $Id: Arrow.hs,v 1.1 2002/02/26 18:19:17 ross Exp $
+-- $Id: Arrow.hs,v 1.2 2002/04/24 17:57:55 ross Exp $
--
-- 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,
+-- /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
---
--- http://www.soi.city.ac.uk/~ross/arrows/
---
------------------------------------------------------------------------------
+-- <http://www.soi.city.ac.uk/~ross/arrows/>.
module Control.Arrow where
-import Prelude
-
import Control.Monad
import Control.Monad.Fix
infixr 1 <<<
-----------------------------------------------------------------------------
--- Arrow classes
+-- * Arrows
+
+-- | 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
- first :: a b c -> a (b,d) (c,d)
- -- The following combinators are placed in the class so that they
- -- can be overridden with more efficient versions if required.
- -- Any replacements should satisfy these equations.
+ -- | 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
- -- Some people prefer the name pure to arr, so both are allowed,
- -- but you must define one of them:
+-- Ordinary functions are arrows.
- pure :: (b -> c) -> a b c
- pure = arr
- arr = pure
+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)
+
+-- | 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))
-----------------------------------------------------------------------------
--- Derived combinators
+-- ** Derived combinators
--- The counterpart of return in arrow notation:
+-- | The identity arrow, which plays the role of 'return' in arrow notation.
returnA :: Arrow a => a b b
returnA = arr id
--- Mirror image of >>>, for a better fit with arrow notation:
+-- | 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
-----------------------------------------------------------------------------
--- Monoid operations
+-- * Monoid operations
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)
+
-----------------------------------------------------------------------------
--- Conditionals
+-- * Conditionals
+
+-- | 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
- left :: a b c -> a (Either b d) (Either c d)
- -- The following combinators are placed in the class so that they
- -- can be overridden with more efficient versions if required.
- -- Any replacements should satisfy these equations.
+ -- | 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)
+
-----------------------------------------------------------------------------
--- Arrow application
+-- * Arrow application
+
+-- | Some arrows allow application of arrow inputs to other inputs.
class Arrow a => ArrowApply a where
app :: a (a b c, b) c
--- Any instance of ArrowApply can be made into an instance if ArrowChoice
--- by defining left = leftApp, where
+instance ArrowApply (->) where
+ app (f,x) = f x
-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
+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 Kliesli arrow (see below), and any instance of ArrowApply defines
--- a monad:
+-- | 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)
arr (\x -> let ArrowMonad h = f x in (h, ())) >>>
app)
------------------------------------------------------------------------------
--- Feedback
-
--- The following operator expresses computations in which a value is
--- recursively defined through the computation, even though the computation
--- occurs only once:
+-- | Any instance of 'ArrowApply' can be made into an instance of
+-- 'ArrowChoice' by defining 'left' = 'leftApp'.
-class Arrow a => ArrowLoop a where
- loop :: a (b,d) (c,d) -> a b c
+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
-----------------------------------------------------------------------------
--- Arrow instances
+-- * Feedback
--- 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)
-
-instance ArrowChoice (->) where
- left f = f +++ id
- right f = id +++ f
- f +++ g = (Left . f) ||| (Right . g)
- (|||) = either
+-- | 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.
-instance ArrowApply (->) where
- app (f,x) = f x
+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
------------------------------------------------------------------------------
--- 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))
-
-instance MonadPlus m => ArrowZero (Kleisli m) where
- zeroArrow = Kleisli (\x -> mzero)
-
-instance MonadPlus m => ArrowPlus (Kleisli m) where
- Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
-
-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)
-
-instance Monad m => ArrowApply (Kleisli m) where
- app = Kleisli (\(Kleisli f, x) -> f x)
-
instance MonadFix m => ArrowLoop (Kleisli m) where
loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
where f' x y = f (x, snd y)
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