1 -----------------------------------------------------------------------------
3 -- Module : Control.Arrow
4 -- Copyright : (c) Ross Paterson 2002
5 -- License : BSD-style (see the LICENSE file in the distribution)
7 -- Maintainer : ross@soi.city.ac.uk
8 -- Stability : experimental
9 -- Portability : portable
11 -- Basic arrow definitions, based on
12 -- /Generalising Monads to Arrows/, by John Hughes,
13 -- /Science of Computer Programming/ 37, pp67-111, May 2000.
14 -- plus a couple of definitions ('returnA' and 'loop') from
15 -- /A New Notation for Arrows/, by Ross Paterson, in /ICFP 2001/,
16 -- Firenze, Italy, pp229-240.
17 -- See these papers for the equations these combinators are expected to
18 -- satisfy. These papers and more information on arrows can be found at
19 -- <http://www.soi.city.ac.uk/~ross/arrows/>.
21 module Control.Arrow where
26 import Control.Monad.Fix
36 -----------------------------------------------------------------------------
39 -- | The basic arrow class.
40 -- Any instance must define either 'arr' or 'pure' (which are synonyms),
41 -- as well as '>>>' and 'first'. The other combinators have sensible
42 -- default definitions, which may be overridden for efficiency.
46 -- | Lift a function to an arrow: you must define either this
48 arr :: (b -> c) -> a b c
51 -- | A synonym for 'arr': you must define one or other of them.
52 pure :: (b -> c) -> a b c
55 -- | Left-to-right composition of arrows.
56 (>>>) :: a b c -> a c d -> a b d
58 -- | Send the first component of the input through the argument
59 -- arrow, and copy the rest unchanged to the output.
60 first :: a b c -> a (b,d) (c,d)
62 -- | A mirror image of 'first'.
64 -- The default definition may be overridden with a more efficient
65 -- version if desired.
66 second :: a b c -> a (d,b) (d,c)
67 second f = arr swap >>> first f >>> arr swap
68 where swap ~(x,y) = (y,x)
70 -- | Split the input between the two argument arrows and combine
71 -- their output. Note that this is in general not a functor.
73 -- The default definition may be overridden with a more efficient
74 -- version if desired.
75 (***) :: a b c -> a b' c' -> a (b,b') (c,c')
76 f *** g = first f >>> second g
78 -- | Fanout: send the input to both argument arrows and combine
81 -- The default definition may be overridden with a more efficient
82 -- version if desired.
83 (&&&) :: a b c -> a b c' -> a b (c,c')
84 f &&& g = arr (\b -> (b,b)) >>> f *** g
86 -- Ordinary functions are arrows.
88 instance Arrow (->) where
93 (f *** g) ~(x,y) = (f x, g y)
95 -- | Kleisli arrows of a monad.
97 newtype Kleisli m a b = Kleisli (a -> m b)
99 instance Monad m => Arrow (Kleisli m) where
100 arr f = Kleisli (return . f)
101 Kleisli f >>> Kleisli g = Kleisli (\b -> f b >>= g)
102 first (Kleisli f) = Kleisli (\ ~(b,d) -> f b >>= \c -> return (c,d))
103 second (Kleisli f) = Kleisli (\ ~(d,b) -> f b >>= \c -> return (d,c))
105 -----------------------------------------------------------------------------
106 -- ** Derived combinators
108 -- | The identity arrow, which plays the role of 'return' in arrow notation.
110 returnA :: Arrow a => a b b
113 -- | Right-to-left composition, for a better fit with arrow notation.
115 (<<<) :: Arrow a => a c d -> a b c -> a b d
118 -----------------------------------------------------------------------------
119 -- * Monoid operations
121 class Arrow a => ArrowZero a where
124 instance MonadPlus m => ArrowZero (Kleisli m) where
125 zeroArrow = Kleisli (\x -> mzero)
127 class ArrowZero a => ArrowPlus a where
128 (<+>) :: a b c -> a b c -> a b c
130 instance MonadPlus m => ArrowPlus (Kleisli m) where
131 Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
133 -----------------------------------------------------------------------------
136 -- | Choice, for arrows that support it. This class underlies the
137 -- [if] and [case] constructs in arrow notation.
138 -- Any instance must define 'left'. The other combinators have sensible
139 -- default definitions, which may be overridden for efficiency.
141 class Arrow a => ArrowChoice a where
143 -- | Feed marked inputs through the argument arrow, passing the
144 -- rest through unchanged to the output.
145 left :: a b c -> a (Either b d) (Either c d)
147 -- | A mirror image of 'left'.
149 -- The default definition may be overridden with a more efficient
150 -- version if desired.
151 right :: a b c -> a (Either d b) (Either d c)
152 right f = arr mirror >>> left f >>> arr mirror
153 where mirror (Left x) = Right x
154 mirror (Right y) = Left y
156 -- | Split the input between the two argument arrows, retagging
157 -- and merging their outputs.
158 -- Note that this is in general not a functor.
160 -- The default definition may be overridden with a more efficient
161 -- version if desired.
162 (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
163 f +++ g = left f >>> right g
165 -- | Fanin: Split the input between the two argument arrows and
166 -- merge their outputs.
168 -- The default definition may be overridden with a more efficient
169 -- version if desired.
170 (|||) :: a b d -> a c d -> a (Either b c) d
171 f ||| g = f +++ g >>> arr untag
172 where untag (Left x) = x
175 instance ArrowChoice (->) where
178 f +++ g = (Left . f) ||| (Right . g)
181 instance Monad m => ArrowChoice (Kleisli m) where
182 left f = f +++ arr id
183 right f = arr id +++ f
184 f +++ g = (f >>> arr Left) ||| (g >>> arr Right)
185 Kleisli f ||| Kleisli g = Kleisli (either f g)
187 -----------------------------------------------------------------------------
188 -- * Arrow application
190 -- | Some arrows allow application of arrow inputs to other inputs.
192 class Arrow a => ArrowApply a where
193 app :: a (a b c, b) c
195 instance ArrowApply (->) where
198 instance Monad m => ArrowApply (Kleisli m) where
199 app = Kleisli (\(Kleisli f, x) -> f x)
201 -- | The 'ArrowApply' class is equivalent to 'Monad': any monad gives rise
202 -- to a 'Kleisli' arrow, and any instance of 'ArrowApply' defines a monad.
204 newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)
206 instance ArrowApply a => Monad (ArrowMonad a) where
207 return x = ArrowMonad (arr (\z -> x))
208 ArrowMonad m >>= f = ArrowMonad (m >>>
209 arr (\x -> let ArrowMonad h = f x in (h, ())) >>>
212 -- | Any instance of 'ArrowApply' can be made into an instance of
213 -- 'ArrowChoice' by defining 'left' = 'leftApp'.
215 leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d)
216 leftApp f = arr ((\b -> (arr (\() -> b) >>> f >>> arr Left, ())) |||
217 (\d -> (arr (\() -> d) >>> arr Right, ()))) >>> app
219 -----------------------------------------------------------------------------
222 -- | The 'loop' operator expresses computations in which an output value is
223 -- fed back as input, even though the computation occurs only once.
224 -- It underlies the [rec] value recursion construct in arrow notation.
226 class Arrow a => ArrowLoop a where
227 loop :: a (b,d) (c,d) -> a b c
229 instance ArrowLoop (->) where
230 loop f b = let (c,d) = f (b,d) in c
232 instance MonadFix m => ArrowLoop (Kleisli m) where
233 loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
234 where f' x y = f (x, snd y)