20e367799f800033782b6043d56cc1fbde54d9ce
[ghc.git] / libraries / base / Control / Arrow.hs
1 -----------------------------------------------------------------------------
2 -- |
3 -- Module : Control.Arrow
4 -- Copyright : (c) Ross Paterson 2002
5 -- License : BSD-style (see the LICENSE file in the distribution)
6 --
7 -- Maintainer : libraries@haskell.org
8 -- Stability : experimental
9 -- Portability : portable
10 --
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.haskell.org/arrows/>.
20
21 module Control.Arrow (
22 -- * Arrows
23 Arrow(..), Kleisli(..),
24 -- ** Derived combinators
25 returnA,
26 (^>>), (>>^),
27 (>>>), (<<<), -- reexported
28 -- ** Right-to-left variants
29 (<<^), (^<<),
30 -- * Monoid operations
31 ArrowZero(..), ArrowPlus(..),
32 -- * Conditionals
33 ArrowChoice(..),
34 -- * Arrow application
35 ArrowApply(..), ArrowMonad(..), leftApp,
36 -- * Feedback
37 ArrowLoop(..)
38 ) where
39
40 import Prelude hiding (id,(.))
41
42 import Control.Monad
43 import Control.Monad.Fix
44 import Control.Category
45
46 infixr 5 <+>
47 infixr 3 ***
48 infixr 3 &&&
49 infixr 2 +++
50 infixr 2 |||
51 infixr 1 ^>>, >>^
52 infixr 1 ^<<, <<^
53
54 -- | The basic arrow class.
55 --
56 -- Minimal complete definition: 'arr' and 'first'.
57 --
58 -- The other combinators have sensible default definitions,
59 -- which may be overridden for efficiency.
60
61 class Category a => Arrow a where
62
63 -- | Lift a function to an arrow.
64 arr :: (b -> c) -> a b c
65
66 -- | Send the first component of the input through the argument
67 -- arrow, and copy the rest unchanged to the output.
68 first :: a b c -> a (b,d) (c,d)
69
70 -- | A mirror image of 'first'.
71 --
72 -- The default definition may be overridden with a more efficient
73 -- version if desired.
74 second :: a b c -> a (d,b) (d,c)
75 second f = arr swap >>> first f >>> arr swap
76 where
77 swap :: (x,y) -> (y,x)
78 swap ~(x,y) = (y,x)
79
80 -- | Split the input between the two argument arrows and combine
81 -- their output. Note that this is in general not a functor.
82 --
83 -- The default definition may be overridden with a more efficient
84 -- version if desired.
85 (***) :: a b c -> a b' c' -> a (b,b') (c,c')
86 f *** g = first f >>> second g
87
88 -- | Fanout: send the input to both argument arrows and combine
89 -- their output.
90 --
91 -- The default definition may be overridden with a more efficient
92 -- version if desired.
93 (&&&) :: a b c -> a b c' -> a b (c,c')
94 f &&& g = arr (\b -> (b,b)) >>> f *** g
95
96 {-# RULES
97 "compose/arr" forall f g .
98 (arr f) . (arr g) = arr (f . g)
99 "first/arr" forall f .
100 first (arr f) = arr (first f)
101 "second/arr" forall f .
102 second (arr f) = arr (second f)
103 "product/arr" forall f g .
104 arr f *** arr g = arr (f *** g)
105 "fanout/arr" forall f g .
106 arr f &&& arr g = arr (f &&& g)
107 "compose/first" forall f g .
108 (first f) . (first g) = first (f . g)
109 "compose/second" forall f g .
110 (second f) . (second g) = second (f . g)
111 #-}
112
113 -- Ordinary functions are arrows.
114
115 instance Arrow (->) where
116 arr f = f
117 first f = f *** id
118 second f = id *** f
119 -- (f *** g) ~(x,y) = (f x, g y)
120 -- sorry, although the above defn is fully H'98, nhc98 can't parse it.
121 (***) f g ~(x,y) = (f x, g y)
122
123 -- | Kleisli arrows of a monad.
124
125 newtype Kleisli m a b = Kleisli { runKleisli :: a -> m b }
126
127 instance Monad m => Category (Kleisli m) where
128 id = Kleisli return
129 (Kleisli f) . (Kleisli g) = Kleisli (\b -> g b >>= f)
130
131 instance Monad m => Arrow (Kleisli m) where
132 arr f = Kleisli (return . f)
133 first (Kleisli f) = Kleisli (\ ~(b,d) -> f b >>= \c -> return (c,d))
134 second (Kleisli f) = Kleisli (\ ~(d,b) -> f b >>= \c -> return (d,c))
135
136 -- | The identity arrow, which plays the role of 'return' in arrow notation.
137
138 returnA :: Arrow a => a b b
139 returnA = arr id
140
141 -- | Precomposition with a pure function.
142 (^>>) :: Arrow a => (b -> c) -> a c d -> a b d
143 f ^>> a = arr f >>> a
144
145 -- | Postcomposition with a pure function.
146 (>>^) :: Arrow a => a b c -> (c -> d) -> a b d
147 a >>^ f = a >>> arr f
148
149 -- | Precomposition with a pure function (right-to-left variant).
150 (<<^) :: Arrow a => a c d -> (b -> c) -> a b d
151 a <<^ f = a <<< arr f
152
153 -- | Postcomposition with a pure function (right-to-left variant).
154 (^<<) :: Arrow a => (c -> d) -> a b c -> a b d
155 f ^<< a = arr f <<< a
156
157 class Arrow a => ArrowZero a where
158 zeroArrow :: a b c
159
160 instance MonadPlus m => ArrowZero (Kleisli m) where
161 zeroArrow = Kleisli (\_ -> mzero)
162
163 class ArrowZero a => ArrowPlus a where
164 (<+>) :: a b c -> a b c -> a b c
165
166 instance MonadPlus m => ArrowPlus (Kleisli m) where
167 Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
168
169 -- | Choice, for arrows that support it. This class underlies the
170 -- @if@ and @case@ constructs in arrow notation.
171 -- Any instance must define 'left'. The other combinators have sensible
172 -- default definitions, which may be overridden for efficiency.
173
174 class Arrow a => ArrowChoice a where
175
176 -- | Feed marked inputs through the argument arrow, passing the
177 -- rest through unchanged to the output.
178 left :: a b c -> a (Either b d) (Either c d)
179
180 -- | A mirror image of 'left'.
181 --
182 -- The default definition may be overridden with a more efficient
183 -- version if desired.
184 right :: a b c -> a (Either d b) (Either d c)
185 right f = arr mirror >>> left f >>> arr mirror
186 where
187 mirror :: Either x y -> Either y x
188 mirror (Left x) = Right x
189 mirror (Right y) = Left y
190
191 -- | Split the input between the two argument arrows, retagging
192 -- and merging their outputs.
193 -- Note that this is in general not a functor.
194 --
195 -- The default definition may be overridden with a more efficient
196 -- version if desired.
197 (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
198 f +++ g = left f >>> right g
199
200 -- | Fanin: Split the input between the two argument arrows and
201 -- merge their outputs.
202 --
203 -- The default definition may be overridden with a more efficient
204 -- version if desired.
205 (|||) :: a b d -> a c d -> a (Either b c) d
206 f ||| g = f +++ g >>> arr untag
207 where
208 untag (Left x) = x
209 untag (Right y) = y
210
211 {-# RULES
212 "left/arr" forall f .
213 left (arr f) = arr (left f)
214 "right/arr" forall f .
215 right (arr f) = arr (right f)
216 "sum/arr" forall f g .
217 arr f +++ arr g = arr (f +++ g)
218 "fanin/arr" forall f g .
219 arr f ||| arr g = arr (f ||| g)
220 "compose/left" forall f g .
221 left f . left g = left (f . g)
222 "compose/right" forall f g .
223 right f . right g = right (f . g)
224 #-}
225
226 instance ArrowChoice (->) where
227 left f = f +++ id
228 right f = id +++ f
229 f +++ g = (Left . f) ||| (Right . g)
230 (|||) = either
231
232 instance Monad m => ArrowChoice (Kleisli m) where
233 left f = f +++ arr id
234 right f = arr id +++ f
235 f +++ g = (f >>> arr Left) ||| (g >>> arr Right)
236 Kleisli f ||| Kleisli g = Kleisli (either f g)
237
238 -- | Some arrows allow application of arrow inputs to other inputs.
239
240 class Arrow a => ArrowApply a where
241 app :: a (a b c, b) c
242
243 instance ArrowApply (->) where
244 app (f,x) = f x
245
246 instance Monad m => ArrowApply (Kleisli m) where
247 app = Kleisli (\(Kleisli f, x) -> f x)
248
249 -- | The 'ArrowApply' class is equivalent to 'Monad': any monad gives rise
250 -- to a 'Kleisli' arrow, and any instance of 'ArrowApply' defines a monad.
251
252 newtype ArrowMonad a b = ArrowMonad (a () b)
253
254 instance ArrowApply a => Monad (ArrowMonad a) where
255 return x = ArrowMonad (arr (\_ -> x))
256 ArrowMonad m >>= f = ArrowMonad $
257 m >>> arr (\x -> let ArrowMonad h = f x in (h, ())) >>> app
258
259 -- | Any instance of 'ArrowApply' can be made into an instance of
260 -- 'ArrowChoice' by defining 'left' = 'leftApp'.
261
262 leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d)
263 leftApp f = arr ((\b -> (arr (\() -> b) >>> f >>> arr Left, ())) |||
264 (\d -> (arr (\() -> d) >>> arr Right, ()))) >>> app
265
266 -- | The 'loop' operator expresses computations in which an output value is
267 -- fed back as input, even though the computation occurs only once.
268 -- It underlies the @rec@ value recursion construct in arrow notation.
269
270 class Arrow a => ArrowLoop a where
271 loop :: a (b,d) (c,d) -> a b c
272
273 instance ArrowLoop (->) where
274 loop f b = let (c,d) = f (b,d) in c
275
276 instance MonadFix m => ArrowLoop (Kleisli m) where
277 loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
278 where f' x y = f (x, snd y)