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
24 import Control.Monad.Fix
34 -----------------------------------------------------------------------------
37 -- | The basic arrow class.
38 -- Any instance must define either 'arr' or 'pure' (which are synonyms),
39 -- as well as '>>>' and 'first'. The other combinators have sensible
40 -- default definitions, which may be overridden for efficiency.
44 -- | Lift a function to an arrow: you must define either this
46 arr :: (b -> c) -> a b c
49 -- | A synonym for 'arr': you must define one or other of them.
50 pure :: (b -> c) -> a b c
53 -- | Left-to-right composition of arrows.
54 (>>>) :: a b c -> a c d -> a b d
56 -- | Send the first component of the input through the argument
57 -- arrow, and copy the rest unchanged to the output.
58 first :: a b c -> a (b,d) (c,d)
60 -- | A mirror image of 'first'.
62 -- The default definition may be overridden with a more efficient
63 -- version if desired.
64 second :: a b c -> a (d,b) (d,c)
65 second f = arr swap >>> first f >>> arr swap
66 where swap ~(x,y) = (y,x)
68 -- | Split the input between the two argument arrows and combine
69 -- their output. Note that this is in general not a functor.
71 -- The default definition may be overridden with a more efficient
72 -- version if desired.
73 (***) :: a b c -> a b' c' -> a (b,b') (c,c')
74 f *** g = first f >>> second g
76 -- | Fanout: send the input to both argument arrows and combine
79 -- The default definition may be overridden with a more efficient
80 -- version if desired.
81 (&&&) :: a b c -> a b c' -> a b (c,c')
82 f &&& g = arr (\b -> (b,b)) >>> f *** g
84 -- Ordinary functions are arrows.
86 instance Arrow (->) where
91 (f *** g) ~(x,y) = (f x, g y)
93 -- | Kleisli arrows of a monad.
95 newtype Kleisli m a b = Kleisli (a -> m b)
97 instance Monad m => Arrow (Kleisli m) where
98 arr f = Kleisli (return . f)
99 Kleisli f >>> Kleisli g = Kleisli (\b -> f b >>= g)
100 first (Kleisli f) = Kleisli (\ ~(b,d) -> f b >>= \c -> return (c,d))
101 second (Kleisli f) = Kleisli (\ ~(d,b) -> f b >>= \c -> return (d,c))
103 -----------------------------------------------------------------------------
104 -- ** Derived combinators
106 -- | The identity arrow, which plays the role of 'return' in arrow notation.
108 returnA :: Arrow a => a b b
111 -- | Right-to-left composition, for a better fit with arrow notation.
113 (<<<) :: Arrow a => a c d -> a b c -> a b d
116 -----------------------------------------------------------------------------
117 -- * Monoid operations
119 class Arrow a => ArrowZero a where
122 instance MonadPlus m => ArrowZero (Kleisli m) where
123 zeroArrow = Kleisli (\x -> mzero)
125 class ArrowZero a => ArrowPlus a where
126 (<+>) :: a b c -> a b c -> a b c
128 instance MonadPlus m => ArrowPlus (Kleisli m) where
129 Kleisli f <+> Kleisli g = Kleisli (\x -> f x `mplus` g x)
131 -----------------------------------------------------------------------------
134 -- | Choice, for arrows that support it. This class underlies the
135 -- [if] and [case] constructs in arrow notation.
136 -- Any instance must define 'left'. The other combinators have sensible
137 -- default definitions, which may be overridden for efficiency.
139 class Arrow a => ArrowChoice a where
141 -- | Feed marked inputs through the argument arrow, passing the
142 -- rest through unchanged to the output.
143 left :: a b c -> a (Either b d) (Either c d)
145 -- | A mirror image of 'left'.
147 -- The default definition may be overridden with a more efficient
148 -- version if desired.
149 right :: a b c -> a (Either d b) (Either d c)
150 right f = arr mirror >>> left f >>> arr mirror
151 where mirror (Left x) = Right x
152 mirror (Right y) = Left y
154 -- | Split the input between the two argument arrows, retagging
155 -- and merging their outputs.
156 -- Note that this is in general not a functor.
158 -- The default definition may be overridden with a more efficient
159 -- version if desired.
160 (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
161 f +++ g = left f >>> right g
163 -- | Fanin: Split the input between the two argument arrows and
164 -- merge their outputs.
166 -- The default definition may be overridden with a more efficient
167 -- version if desired.
168 (|||) :: a b d -> a c d -> a (Either b c) d
169 f ||| g = f +++ g >>> arr untag
170 where untag (Left x) = x
173 instance ArrowChoice (->) where
176 f +++ g = (Left . f) ||| (Right . g)
179 instance Monad m => ArrowChoice (Kleisli m) where
180 left f = f +++ arr id
181 right f = arr id +++ f
182 f +++ g = (f >>> arr Left) ||| (g >>> arr Right)
183 Kleisli f ||| Kleisli g = Kleisli (either f g)
185 -----------------------------------------------------------------------------
186 -- * Arrow application
188 -- | Some arrows allow application of arrow inputs to other inputs.
190 class Arrow a => ArrowApply a where
191 app :: a (a b c, b) c
193 instance ArrowApply (->) where
196 instance Monad m => ArrowApply (Kleisli m) where
197 app = Kleisli (\(Kleisli f, x) -> f x)
199 -- | The 'ArrowApply' class is equivalent to 'Monad': any monad gives rise
200 -- to a 'Kleisli' arrow, and any instance of 'ArrowApply' defines a monad.
202 newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)
204 instance ArrowApply a => Monad (ArrowMonad a) where
205 return x = ArrowMonad (arr (\z -> x))
206 ArrowMonad m >>= f = ArrowMonad (m >>>
207 arr (\x -> let ArrowMonad h = f x in (h, ())) >>>
210 -- | Any instance of 'ArrowApply' can be made into an instance of
211 -- 'ArrowChoice' by defining 'left' = 'leftApp'.
213 leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d)
214 leftApp f = arr ((\b -> (arr (\() -> b) >>> f >>> arr Left, ())) |||
215 (\d -> (arr (\() -> d) >>> arr Right, ()))) >>> app
217 -----------------------------------------------------------------------------
220 -- | The 'loop' operator expresses computations in which an output value is
221 -- fed back as input, even though the computation occurs only once.
222 -- It underlies the [rec] value recursion construct in arrow notation.
224 class Arrow a => ArrowLoop a where
225 loop :: a (b,d) (c,d) -> a b c
227 instance ArrowLoop (->) where
228 loop f b = let (c,d) = f (b,d) in c
230 instance MonadFix m => ArrowLoop (Kleisli m) where
231 loop (Kleisli f) = Kleisli (liftM fst . mfix . f')
232 where f' x y = f (x, snd y)