Copyright	(c) Ross Paterson 2002
License	BSD-style (see the LICENSE file in the distribution)
Maintainer	libraries@haskell.org
Stability	provisional
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell2010

Control.Arrow

Contents

Arrows
- Derived combinators
- Right-to-left variants
Monoid operations
Conditionals
Arrow application
Feedback

Description

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, Firenze, Italy, pp229-240.

These papers and more information on arrows can be found at http://www.haskell.org/arrows/.

Synopsis

Arrows

class Category a => Arrow a where #

The basic arrow class.

Instances should satisfy the following laws:

```
arr id = id
```
```
arr (f >>> g) = arr f >>> arr g
```
```
first (arr f) = arr (first f)
```
```
first (f >>> g) = first f >>> first g
```
```
first f >>> arr fst = arr fst >>> f
```

first f >>> arr (id *** g) = arr (id *** g) >>> first f

first (first f) >>> arr assoc = arr assoc >>> first f

where

assoc ((a,b),c) = (a,(b,c))

The other combinators have sensible default definitions, which may be overridden for efficiency.

Minimal complete definition

arr, (first | (***))

Methods

arr :: (b -> c) -> a b c #

Lift a function to an arrow.

first :: a b c -> a (b, d) (c, d) #

Send the first component of the input through the argument arrow, and copy the rest unchanged to the output.

second :: a b c -> a (d, b) (d, c) #

A mirror image of first.

The default definition may be overridden with a more efficient version if desired.

(***) :: a b c -> a b' c' -> a (b, b') (c, c') infixr 3 #

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 (c, c') infixr 3 #

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.

Instances

Arrow (->) #
Methods arr :: (b -> c) -> b -> c # first :: (b -> c) -> (b, d) -> (c, d) # second :: (b -> c) -> (d, b) -> (d, c) # (***) :: (b -> c) -> (b' -> c') -> (b, b') -> (c, c') # (&&&) :: (b -> c) -> (b -> c') -> b -> (c, c') #
Monad m => Arrow (Kleisli m) #
Methods arr :: (b -> c) -> Kleisli m b c # first :: Kleisli m b c -> Kleisli m (b, d) (c, d) # second :: Kleisli m b c -> Kleisli m (d, b) (d, c) # (***) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (b, b') (c, c') # (&&&) :: Kleisli m b c -> Kleisli m b c' -> Kleisli m b (c, c') #

newtype Kleisli m a b #

Kleisli arrows of a monad.

Constructors

Kleisli
Fields runKleisli :: a -> m b

Instances

MonadFix m => ArrowLoop (Kleisli m) #	Beware that for many monads (those for which the `>>=` operation is strict) this instance will not satisfy the right-tightening law required by the `ArrowLoop` class.
Methods loop :: Kleisli m (b, d) (c, d) -> Kleisli m b c #
Monad m => ArrowApply (Kleisli m) #
Methods app :: Kleisli m (Kleisli m b c, b) c #
Monad m => ArrowChoice (Kleisli m) #
Methods left :: Kleisli m b c -> Kleisli m (Either b d) (Either c d) # right :: Kleisli m b c -> Kleisli m (Either d b) (Either d c) # (+++) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (Either b b') (Either c c') # (\|\|\|) :: Kleisli m b d -> Kleisli m c d -> Kleisli m (Either b c) d #
MonadPlus m => ArrowPlus (Kleisli m) #
Methods (<+>) :: Kleisli m b c -> Kleisli m b c -> Kleisli m b c #
MonadPlus m => ArrowZero (Kleisli m) #
Methods zeroArrow :: Kleisli m b c #
Monad m => Arrow (Kleisli m) #
Methods arr :: (b -> c) -> Kleisli m b c # first :: Kleisli m b c -> Kleisli m (b, d) (c, d) # second :: Kleisli m b c -> Kleisli m (d, b) (d, c) # (***) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (b, b') (c, c') # (&&&) :: Kleisli m b c -> Kleisli m b c' -> Kleisli m b (c, c') #
Monad m => Category * (Kleisli m) #
Methods id :: cat a a # (.) :: cat b c -> cat a b -> cat a c #

Derived combinators

returnA :: Arrow a => a b b #

The identity arrow, which plays the role of return in arrow notation.

(^>>) :: Arrow a => (b -> c) -> a c d -> a b d infixr 1 #

Precomposition with a pure function.

(>>^) :: Arrow a => a b c -> (c -> d) -> a b d infixr 1 #

Postcomposition with a pure function.

(>>>) :: Category cat => cat a b -> cat b c -> cat a c infixr 1 #

Left-to-right composition

(<<<) :: Category cat => cat b c -> cat a b -> cat a c infixr 1 #

Right-to-left composition

Right-to-left variants

(<<^) :: Arrow a => a c d -> (b -> c) -> a b d infixr 1 #

Precomposition with a pure function (right-to-left variant).

(^<<) :: Arrow a => (c -> d) -> a b c -> a b d infixr 1 #

Postcomposition with a pure function (right-to-left variant).

Monoid operations

class Arrow a => ArrowZero a where #

Minimal complete definition

zeroArrow

Methods

zeroArrow :: a b c #

Instances

MonadPlus m => ArrowZero (Kleisli m) #
Methods zeroArrow :: Kleisli m b c #

class ArrowZero a => ArrowPlus a where #

A monoid on arrows.

Minimal complete definition

(<+>)

Methods

(<+>) :: a b c -> a b c -> a b c infixr 5 #

An associative operation with identity zeroArrow.

Instances

MonadPlus m => ArrowPlus (Kleisli m) #
Methods (<+>) :: Kleisli m b c -> Kleisli m b c -> Kleisli m b c #

Conditionals

class Arrow a => ArrowChoice a where #

Choice, for arrows that support it. This class underlies the if and case constructs in arrow notation.

Instances should satisfy the following laws:

```
left (arr f) = arr (left f)
```
```
left (f >>> g) = left f >>> left g
```
```
f >>> arr Left = arr Left >>> left f
```

left f >>> arr (id +++ g) = arr (id +++ g) >>> left f

left (left f) >>> arr assocsum = arr assocsum >>> left f

where

assocsum (Left (Left x)) = Left x
assocsum (Left (Right y)) = Right (Left y)
assocsum (Right z) = Right (Right z)

The other combinators have sensible default definitions, which may be overridden for efficiency.

Minimal complete definition

left | (+++)

Methods

left :: a b c -> a (Either b d) (Either c d) #

Feed marked inputs through the argument arrow, passing the rest through unchanged to the output.

right :: a b c -> a (Either d b) (Either d c) #

A mirror image of left.

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') infixr 2 #

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 d -> a c d -> a (Either b c) d infixr 2 #

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.

Instances

ArrowChoice (->) #
Methods left :: (b -> c) -> Either b d -> Either c d # right :: (b -> c) -> Either d b -> Either d c # (+++) :: (b -> c) -> (b' -> c') -> Either b b' -> Either c c' # (\|\|\|) :: (b -> d) -> (c -> d) -> Either b c -> d #
Monad m => ArrowChoice (Kleisli m) #
Methods left :: Kleisli m b c -> Kleisli m (Either b d) (Either c d) # right :: Kleisli m b c -> Kleisli m (Either d b) (Either d c) # (+++) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (Either b b') (Either c c') # (\|\|\|) :: Kleisli m b d -> Kleisli m c d -> Kleisli m (Either b c) d #

Arrow application

class Arrow a => ArrowApply a where #

Some arrows allow application of arrow inputs to other inputs. Instances should satisfy the following laws:

first (arr (\x -> arr (\y -> (x,y)))) >>> app = id

first (arr (g >>>)) >>> app = second g >>> app

```
first (arr (>>> h)) >>> app = app >>> h
```

Such arrows are equivalent to monads (see ArrowMonad).

Minimal complete definition

app

Methods

app :: a (a b c, b) c #

Instances

ArrowApply (->) #
Methods app :: (b -> c, b) -> c #
Monad m => ArrowApply (Kleisli m) #
Methods app :: Kleisli m (Kleisli m b c, b) c #

newtype ArrowMonad a b #

The ArrowApply class is equivalent to Monad: any monad gives rise to a Kleisli arrow, and any instance of ArrowApply defines a monad.

Constructors

ArrowMonad (a () b)

Instances

ArrowApply a => Monad (ArrowMonad a) #
Methods (>>=) :: ArrowMonad a a -> (a -> ArrowMonad a b) -> ArrowMonad a b # (>>) :: ArrowMonad a a -> ArrowMonad a b -> ArrowMonad a b # return :: a -> ArrowMonad a a # fail :: String -> ArrowMonad a a #
Arrow a => Functor (ArrowMonad a) #
Methods fmap :: (a -> b) -> ArrowMonad a a -> ArrowMonad a b # (<$) :: a -> ArrowMonad a b -> ArrowMonad a a #
Arrow a => Applicative (ArrowMonad a) #
Methods pure :: a -> ArrowMonad a a # (<>) :: ArrowMonad a (a -> b) -> ArrowMonad a a -> ArrowMonad a b # (>) :: ArrowMonad a a -> ArrowMonad a b -> ArrowMonad a b # (<*) :: ArrowMonad a a -> ArrowMonad a b -> ArrowMonad a a #
(ArrowApply a, ArrowPlus a) => MonadPlus (ArrowMonad a) #
Methods mzero :: ArrowMonad a a # mplus :: ArrowMonad a a -> ArrowMonad a a -> ArrowMonad a a #
ArrowPlus a => Alternative (ArrowMonad a) #
Methods empty :: ArrowMonad a a # (<\|>) :: ArrowMonad a a -> ArrowMonad a a -> ArrowMonad a a # some :: ArrowMonad a a -> ArrowMonad a [a] # many :: ArrowMonad a a -> ArrowMonad a [a] #

leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d) #

Any instance of ArrowApply can be made into an instance of ArrowChoice by defining left = leftApp.

Feedback

class Arrow a => ArrowLoop a where #

The loop operator expresses computations in which an output value is fed back as input, although the computation occurs only once. It underlies the rec value recursion construct in arrow notation. loop should satisfy the following laws:

extension: loop (arr f) = arr (\ b -> fst (fix (\ (c,d) -> f (b,d))))
left tightening: loop (first h >>> f) = h >>> loop f
right tightening: loop (f >>> first h) = loop f >>> h
sliding: loop (f >>> arr (id *** k)) = loop (arr (id *** k) >>> f)
vanishing: loop (loop f) = loop (arr unassoc >>> f >>> arr assoc)
superposing: second (loop f) = loop (arr assoc >>> second f >>> arr unassoc)

where

assoc ((a,b),c) = (a,(b,c))
unassoc (a,(b,c)) = ((a,b),c)

Minimal complete definition

loop

Methods

loop :: a (b, d) (c, d) -> a b c #

Instances

ArrowLoop (->) #
Methods loop :: ((b, d) -> (c, d)) -> b -> c #
MonadFix m => ArrowLoop (Kleisli m) #	Beware that for many monads (those for which the `>>=` operation is strict) this instance will not satisfy the right-tightening law required by the `ArrowLoop` class.
Methods loop :: Kleisli m (b, d) (c, d) -> Kleisli m b c #