base-4.7.0.2: Basic libraries

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

- class Category a => Arrow a where
  - arr :: (b -> c) -> a b c
  - first :: a b c -> a (b, d) (c, d)
  - second :: a b c -> a (d, b) (d, c)
  - (***) :: a b c -> a b' c' -> a (b, b') (c, c')
  - (&&&) :: a b c -> a b c' -> a b (c, c')
- newtype Kleisli m a b = Kleisli {
  - runKleisli :: a -> m b
  }
- returnA :: Arrow a => a b b
- (^>>) :: Arrow a => (b -> c) -> a c d -> a b d
- (>>^) :: Arrow a => a b c -> (c -> d) -> a b d
- (>>>) :: Category cat => cat a b -> cat b c -> cat a c
- (<<<) :: Category cat => cat b c -> cat a b -> cat a c
- (<<^) :: Arrow a => a c d -> (b -> c) -> a b d
- (^<<) :: Arrow a => (c -> d) -> a b c -> a b d
- class Arrow a => ArrowZero a where
  - zeroArrow :: a b c
- class ArrowZero a => ArrowPlus a where
  - (<+>) :: a b c -> a b c -> a b c
- class Arrow a => ArrowChoice a where
  - left :: a b c -> a (Either b d) (Either c d)
  - right :: a b c -> a (Either d b) (Either d c)
  - (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')
  - (|||) :: a b d -> a c d -> a (Either b c) d
- class Arrow a => ArrowApply a where
  - app :: a (a b c, b) c
- newtype ArrowMonad a b = ArrowMonad (a () b)
- leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d)
- class Arrow a => ArrowLoop a where
  - loop :: a (b, d) (c, d) -> a b c

# Arrows

class Category a => Arrow a where

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

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

newtype Kleisli m a b

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

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

Methods

zeroArrow :: a b c

Instances

MonadPlus m => ArrowZero (Kleisli m) |

class ArrowZero a => ArrowPlus a where

A monoid on arrows.

Methods

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

An associative operation with identity `zeroArrow`.

Instances

MonadPlus m => ArrowPlus (Kleisli m) |

# Conditionals

class Arrow a => ArrowChoice a where

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)

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

# 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`).

Methods

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

Instances

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

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

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)

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)

Methods

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

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.