[project @ 2003-05-12 10:16:22 by ross]
[packages/old-time.git] / 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 : ross@soi.city.ac.uk
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 -- * Monoid operations
27 ArrowZero(..), ArrowPlus(..),
28 -- * Conditionals
29 ArrowChoice(..),
30 -- * Arrow application
31 ArrowApply(..), ArrowMonad(..), leftApp,
32 -- * Feedback
33 ArrowLoop(..)
34 ) where
35
36 import Prelude
37
38 import Control.Monad
39 import Control.Monad.Fix
40
41 infixr 5 <+>
42 infixr 3 ***
43 infixr 3 &&&
44 infixr 2 +++
45 infixr 2 |||
46 infixr 1 >>>
47 infixr 1 <<<
48
49 -- | The basic arrow class.
50 -- Any instance must define either 'arr' or 'pure' (which are synonyms),
51 -- as well as '>>>' and 'first'. The other combinators have sensible
52 -- default definitions, which may be overridden for efficiency.
53
54 class Arrow a where
55
56 -- | Lift a function to an arrow: you must define either this
57 -- or 'pure'.
58 arr :: (b -> c) -> a b c
59 arr = pure
60
61 -- | A synonym for 'arr': you must define one or other of them.
62 pure :: (b -> c) -> a b c
63 pure = arr
64
65 -- | Left-to-right composition of arrows.
66 (>>>) :: a b c -> a c d -> a b d
67
68 -- | Send the first component of the input through the argument
69 -- arrow, and copy the rest unchanged to the output.
70 first :: a b c -> a (b,d) (c,d)
71
72 -- | A mirror image of 'first'.
73 --
74 -- The default definition may be overridden with a more efficient
75 -- version if desired.
76 second :: a b c -> a (d,b) (d,c)
77 second f = arr swap >>> first f >>> arr swap
78 where 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 -- Ordinary functions are arrows.
97
98 instance Arrow (->) where
99 arr f = f
100 f >>> g = g . f
101 first f = f *** id
102 second f = id *** f
103 -- (f *** g) ~(x,y) = (f x, g y)
104 -- sorry, although the above defn is fully H'98, nhc98 can't parse it.
105 (***) f g ~(x,y) = (f x, g y)
106
107 -- | Kleisli arrows of a monad.
108
109 newtype Kleisli m a b = Kleisli (a -> m b)
110
111 instance Monad m => Arrow (Kleisli m) where
112 arr f = Kleisli (return . f)
113 Kleisli f >>> Kleisli g = Kleisli (\b -> f b >>= g)
114 first (Kleisli f) = Kleisli (\ ~(b,d) -> f b >>= \c -> return (c,d))
115 second (Kleisli f) = Kleisli (\ ~(d,b) -> f b >>= \c -> return (d,c))
116
117 -- | The identity arrow, which plays the role of 'return' in arrow notation.
118
119 returnA :: Arrow a => a b b
120 returnA = arr id
121
122 -- | Right-to-left composition, for a better fit with arrow notation.
123
124 (<<<) :: Arrow a => a c d -> a b c -> a b d
125 f <<< g = g >>> f
126
127 class Arrow a => ArrowZero a where
128 zeroArrow :: a b c
129
130 instance MonadPlus m => ArrowZero (Kleisli m) where
131 zeroArrow = Kleisli (\x -> mzero)
132
133 class ArrowZero a => ArrowPlus a where
134 (<+>) :: a b c -> a b c -> a b c
135
136 instance MonadPlus m => ArrowPlus (Kleisli m) where
137 Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
138
139 -- | Choice, for arrows that support it. This class underlies the
140 -- @if@ and @case@ constructs in arrow notation.
141 -- Any instance must define 'left'. The other combinators have sensible
142 -- default definitions, which may be overridden for efficiency.
143
144 class Arrow a => ArrowChoice a where
145
146 -- | Feed marked inputs through the argument arrow, passing the
147 -- rest through unchanged to the output.
148 left :: a b c -> a (Either b d) (Either c d)
149
150 -- | A mirror image of 'left'.
151 --
152 -- The default definition may be overridden with a more efficient
153 -- version if desired.
154 right :: a b c -> a (Either d b) (Either d c)
155 right f = arr mirror >>> left f >>> arr mirror
156 where mirror (Left x) = Right x
157 mirror (Right y) = Left y
158
159 -- | Split the input between the two argument arrows, retagging
160 -- and merging their outputs.
161 -- Note that this is in general not a functor.
162 --
163 -- The default definition may be overridden with a more efficient
164 -- version if desired.
165 (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
166 f +++ g = left f >>> right g
167
168 -- | Fanin: Split the input between the two argument arrows and
169 -- merge their outputs.
170 --
171 -- The default definition may be overridden with a more efficient
172 -- version if desired.
173 (|||) :: a b d -> a c d -> a (Either b c) d
174 f ||| g = f +++ g >>> arr untag
175 where untag (Left x) = x
176 untag (Right y) = y
177
178 instance ArrowChoice (->) where
179 left f = f +++ id
180 right f = id +++ f
181 f +++ g = (Left . f) ||| (Right . g)
182 (|||) = either
183
184 instance Monad m => ArrowChoice (Kleisli m) where
185 left f = f +++ arr id
186 right f = arr id +++ f
187 f +++ g = (f >>> arr Left) ||| (g >>> arr Right)
188 Kleisli f ||| Kleisli g = Kleisli (either f g)
189
190 -- | Some arrows allow application of arrow inputs to other inputs.
191
192 class Arrow a => ArrowApply a where
193 app :: a (a b c, b) c
194
195 instance ArrowApply (->) where
196 app (f,x) = f x
197
198 instance Monad m => ArrowApply (Kleisli m) where
199 app = Kleisli (\(Kleisli f, x) -> f x)
200
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.
203
204 newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)
205
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, ())) >>>
210 app)
211
212 -- | Any instance of 'ArrowApply' can be made into an instance of
213 -- 'ArrowChoice' by defining 'left' = 'leftApp'.
214
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
218
219 -- | The 'loop' operator expresses computations in which an output value is
220 -- fed back as input, even though the computation occurs only once.
221 -- It underlies the @rec@ value recursion construct in arrow notation.
222
223 class Arrow a => ArrowLoop a where
224 loop :: a (b,d) (c,d) -> a b c
225
226 instance ArrowLoop (->) where
227 loop f b = let (c,d) = f (b,d) in c
228
229 instance MonadFix m => ArrowLoop (Kleisli m) where
230 loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
231 where f' x y = f (x, snd y)