base-4.8.1.0: Basic libraries

Control.Arrow

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.

Synopsis

# Arrows

class Category a => Arrow a where Source

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

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 (->) Source Methodsarr :: (b -> c) -> (->) b c Sourcefirst :: (->) b c -> (->) (b, d) (c, d) Sourcesecond :: (->) b c -> (->) (d, b) (d, c) Source(***) :: (->) b c -> (->) b' c' -> (->) (b, b') (c, c') Source(&&&) :: (->) b c -> (->) b c' -> (->) b (c, c') Source Monad m => Arrow (Kleisli m) Source Methodsarr :: (b -> c) -> Kleisli m b c Sourcefirst :: Kleisli m b c -> Kleisli m (b, d) (c, d) Sourcesecond :: Kleisli m b c -> Kleisli m (d, b) (d, c) Source(***) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (b, b') (c, c') Source(&&&) :: Kleisli m b c -> Kleisli m b c' -> Kleisli m b (c, c') Source

newtype Kleisli m a b Source

Constructors

 Kleisli FieldsrunKleisli :: a -> m b

Instances

 Monad m => Category * (Kleisli m) Source Methodsid :: Kleisli m a a Source(.) :: Kleisli m b c -> Kleisli m a b -> Kleisli m a c Source MonadFix m => ArrowLoop (Kleisli m) Source 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. Methodsloop :: Kleisli m (b, d) (c, d) -> Kleisli m b c Source Monad m => ArrowApply (Kleisli m) Source Methodsapp :: Kleisli m (Kleisli m b c, b) c Source Monad m => ArrowChoice (Kleisli m) Source Methodsleft :: Kleisli m b c -> Kleisli m (Either b d) (Either c d) Sourceright :: Kleisli m b c -> Kleisli m (Either d b) (Either d c) Source(+++) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (Either b b') (Either c c') Source(|||) :: Kleisli m b d -> Kleisli m c d -> Kleisli m (Either b c) d Source MonadPlus m => ArrowPlus (Kleisli m) Source Methods(<+>) :: Kleisli m b c -> Kleisli m b c -> Kleisli m b c Source MonadPlus m => ArrowZero (Kleisli m) Source MethodszeroArrow :: Kleisli m b c Source Monad m => Arrow (Kleisli m) Source Methodsarr :: (b -> c) -> Kleisli m b c Sourcefirst :: Kleisli m b c -> Kleisli m (b, d) (c, d) Sourcesecond :: Kleisli m b c -> Kleisli m (d, b) (d, c) Source(***) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (b, b') (c, c') Source(&&&) :: Kleisli m b c -> Kleisli m b c' -> Kleisli m b (c, c') Source

## 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) Source MethodszeroArrow :: Kleisli m b c Source

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) Source Methods(<+>) :: Kleisli m b c -> Kleisli m b c -> Kleisli m b c Source

# 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.

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) 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 (->) Source Methodsleft :: (->) b c -> (->) (Either b d) (Either c d) Sourceright :: (->) b c -> (->) (Either d b) (Either d c) Source(+++) :: (->) b c -> (->) b' c' -> (->) (Either b b') (Either c c') Source(|||) :: (->) b d -> (->) c d -> (->) (Either b c) d Source Monad m => ArrowChoice (Kleisli m) Source Methodsleft :: Kleisli m b c -> Kleisli m (Either b d) (Either c d) Sourceright :: Kleisli m b c -> Kleisli m (Either d b) (Either d c) Source(+++) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (Either b b') (Either c c') Source(|||) :: Kleisli m b d -> Kleisli m c d -> Kleisli m (Either b c) d Source

# 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 (->) Source Methodsapp :: (->) ((->) b c, b) c Source Monad m => ArrowApply (Kleisli m) Source Methodsapp :: Kleisli m (Kleisli m b c, b) c 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

Instances

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 (->) Source Methodsloop :: (->) (b, d) (c, d) -> (->) b c Source MonadFix m => ArrowLoop (Kleisli m) Source 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. Methodsloop :: Kleisli m (b, d) (c, d) -> Kleisli m b c Source