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 Source

The basic arrow class.

Minimal complete definition: arr and first, satisfying the 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 Source

Lift a function to an arrow.

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

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) Source

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 Source

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 Source

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 (->)
Monad m => Arrow (Kleisli m)

newtype Kleisli m a b Source

Kleisli arrows of a monad.

Constructors

Kleisli
Fields runKleisli :: a -> m b

Instances

Monad m => Category * (Kleisli m)
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.
Monad m => ArrowApply (Kleisli m)
Monad m => ArrowChoice (Kleisli m)
MonadPlus m => ArrowPlus (Kleisli m)
MonadPlus m => ArrowZero (Kleisli m)
Monad m => Arrow (Kleisli m)

Derived combinators

returnA :: Arrow a => a b b Source

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

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

Precomposition with a pure function.

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

Postcomposition with a pure function.

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

Left-to-right composition

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

Right-to-left composition

Right-to-left variants

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

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

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

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

Monoid operations

class Arrow a => ArrowZero a where Source

Methods

zeroArrow :: a b c Source

Instances

MonadPlus m => ArrowZero (Kleisli m)

class ArrowZero a => ArrowPlus a where Source

A monoid on arrows.

Methods

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

An associative operation with identity zeroArrow.

Instances

MonadPlus m => ArrowPlus (Kleisli m)

Conditionals

class Arrow a => ArrowChoice a where Source

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

Minimal complete definition: left, satisfying the 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) Source

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) Source

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 Source

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 Source

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 (->)
Monad m => ArrowChoice (Kleisli m)

Arrow application

class Arrow a => ArrowApply a where Source

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).

Methods

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

Instances

ArrowApply (->)
Monad m => ArrowApply (Kleisli m)

newtype ArrowMonad a b Source

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

ArrowPlus a => Alternative (ArrowMonad a)
ArrowApply a => Monad (ArrowMonad a)
Arrow a => Functor (ArrowMonad a)
(ArrowApply a, ArrowPlus a) => MonadPlus (ArrowMonad a)
Arrow a => Applicative (ArrowMonad a)

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

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

Feedback

class Arrow a => ArrowLoop a where Source

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)

Methods

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

Instances

ArrowLoop (->)
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.