# Arrow tutorial

### From HaskellWiki

(Added a definition of 'id' to 'instance Category SimpleFunc') |
(The example I added may not be helpful; removed it.) |
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<haskell> | <haskell> | ||

− | f &&& g = split >>> first f >>> second g | + | f &&& g = split >>> first f >>> second g |

− | + | -- = split >>> f *** g | |

</haskell> | </haskell> | ||

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> liftA2 :: (Arrow a) => (b -> c -> d) -> a e b -> a e c -> a e d | > liftA2 :: (Arrow a) => (b -> c -> d) -> a e b -> a e c -> a e d | ||

> liftA2 op f g = split >>> first f >>> second g >>> unsplit op | > liftA2 op f g = split >>> first f >>> second g >>> unsplit op | ||

− | > | + | > -- = f &&& g >>> unsplit op |

</haskell> | </haskell> | ||

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(4, 25) -> 29 applies (+) to tuple elements. | (4, 25) -> 29 applies (+) to tuple elements. | ||

− | +------> f | + | +------> f ---------+ |

− | | | + | | v |

− | 8 ---> (split) ---> | + | 8 ---> (split) (unsplit (+)) ----> 29 |

+ | | ^ | ||

+ | +------> g ---------+ | ||

so we see that h is a new arrow that when applied to 8, applies 8 to f | so we see that h is a new arrow that when applied to 8, applies 8 to f | ||

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h wasn't defined as a linear combination of arrows. GHC has | h wasn't defined as a linear combination of arrows. GHC has | ||

a do-notation that simplifies this in a similar way to how | a do-notation that simplifies this in a similar way to how | ||

− | do-notation simplifies monadic computation. | + | do-notation simplifies monadic computation. The h function |

− | + | ||

can be defined as: | can be defined as: | ||

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</haskell> | </haskell> | ||

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== Kleisli Arrows == | == Kleisli Arrows == | ||

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* Original version - Nov 19, 2006, Tim Newsham. | * Original version - Nov 19, 2006, Tim Newsham. | ||

+ | \ |

## Latest revision as of 18:05, 15 July 2014

> {-# LANGUAGE Arrows #-} > module ArrowFun where > import Control.Arrow > import Control.Category > import Prelude hiding (id,(.))

## Contents |

## [edit] 1 The Arrow

Arrow a b c represents a process that takes as input something of type b and outputs something of type c.

Arr builds an arrow out of a function. This function is arrow-specific. Its signature is

arr :: (Arrow a) => (b -> c) -> a b c

Arrow composition is achieved with (>>>). This takes two arrows and chains them together, one after another. It is also arrow- specific. Its signature is:

(>>>) :: (Arrow a) => a b c -> a c d -> a b d

First and second make a new arrow out of an existing arrow. They perform a transformation (given by their argument) on either the first or the second item of a pair. These definitions are arrow-specific. Their signatures are:

first :: (Arrow a) => a b c -> a (b, d) (c, d) second :: (Arrow a) => a b c -> a (d, b) (d, c)

First and second may seem pretty strange at first, but they'll make sense in a few minutes.

That's it for the arrow-specific definitions.

## [edit] 2 A Simple Arrow

Let's define a really simple arrow as an example. Our simple arrow is just a function mapping an input to an output. We don't really need arrows for something this simple, but we could use something this simple to explain arrows.

> newtype SimpleFunc a b = SimpleFunc { > runF :: (a -> b) > } > > instance Arrow SimpleFunc where > arr f = SimpleFunc f > first (SimpleFunc f) = SimpleFunc (mapFst f) > where mapFst g (a,b) = (g a, b) > second (SimpleFunc f) = SimpleFunc (mapSnd f) > where mapSnd g (a,b) = (a, g b) > > instance Category SimpleFunc where > (SimpleFunc g) . (SimpleFunc f) = SimpleFunc (g . f) > id = arr id

## [edit] 3 Some Arrow Operations

Now lets define some operations that are generic to all arrows.

Split is an arrow that splits a single value into a pair of duplicate values:

> split :: (Arrow a) => a b (b, b) > split = arr (\x -> (x,x))

Unsplit is an arrow that takes a pair of values and combines them to return a single value:

> unsplit :: (Arrow a) => (b -> c -> d) -> a (b, c) d > unsplit = arr . uncurry > -- arr (\op (x,y) -> x `op` y)

(***) combines two arrows into a new arrow by running the two arrows on a pair of values (one arrow on the first item of the pair and one arrow on the second item of the pair).

f *** g = first f >>> second g

(&&&) combines two arrows into a new arrow by running the two arrows on the same value:

f &&& g = split >>> first f >>> second g -- = split >>> f *** g

LiftA2 makes a new arrow that combines the output from two arrows using a binary operation. It works by splitting a value and operating on both halfs and then combining the result:

> liftA2 :: (Arrow a) => (b -> c -> d) -> a e b -> a e c -> a e d > liftA2 op f g = split >>> first f >>> second g >>> unsplit op > -- = f &&& g >>> unsplit op

## [edit] 4 An Example

Now let's build something using our simple arrow definition and some of the tools we just created. We start with two simple arrows, f and g. F halves its input and g triples its input and adds one:

> f, g :: SimpleFunc Int Int > f = arr (`div` 2) > g = arr (\x -> x*3 + 1)

We can combine these together using liftA2:

> h :: SimpleFunc Int Int > h = liftA2 (+) f g > > hOutput :: Int > hOutput = runF h 8

What is h? How does it work? The process defined by h is (split >>> first f >>> second g >>> unsplit (+)). Lets work through an application of h to some value, 8:

8 -> (8, 8) split (8, 8) -> (4, 8) first f (x `div` 2 of the first element) (4, 8) -> (4, 25) second g (3*x + 1 of the second element) (4, 25) -> 29 applies (+) to tuple elements.

+------> f ---------+ | v 8 ---> (split) (unsplit (+)) ----> 29 | ^ +------> g ---------+

so we see that h is a new arrow that when applied to 8, applies 8 to f and applies 8 to g and adds the results.

A lot of juggling occurred to get the plumbing right since h wasn't defined as a linear combination of arrows. GHC has a do-notation that simplifies this in a similar way to how do-notation simplifies monadic computation. The h function can be defined as:

> h' :: SimpleFunc Int Int > h' = proc x -> do > fx <- f -< x > gx <- g -< x > returnA -< (fx + gx) > > hOutput' :: Int > hOutput' = runF h' 8

## [edit] 5 Kleisli Arrows

Let's move on to something a little fancier now: Kleisli arrows. A Kleisli arrow (Kleisli m a b) is the arrow (a -> m b) for all monads. It's defined in Control.Arrows similarly to our SimpleFunc:

newtype Kleisli m a b = Kleisli { runKleisli :: (a -> m b) }

It comes complete with its own definitions for arr, first, second and (>>>). This means that all multi-value functions (a -> [b]) are already defined as Kleisli arrows (because [] is a monad)! (>>>) performs composition, keeping track of all the multiple results. Split, (&&&) and (***) are all defined as before. So for example:

> plusminus, double, h2 :: Kleisli [] Int Int > plusminus = Kleisli (\x -> [x, -x]) > double = arr (* 2) > h2 = liftA2 (+) plusminus double > > h2Output :: [Int] > h2Output = runKleisli h2 8

## [edit] 6 A Teaser

Finally, here is a little teaser. There is an arrow function called returnA which returns an identity arrow. There is an ArrowPlus class that includes a zeroArrow (which for the list monad is an arrow that always returns the empty list) and a <+> operator (which takes the results from two arrows and concatenates them). We can build up some pretty interesting string transformations (the multi-valued function String -> [String]) using Kleisli arrows:

> main :: IO () > main = do > let > prepend x = arr (x ++) > append x = arr (++ x) > withId t = returnA <+> t > xform = (withId $ prepend "<") >>> > (withId $ append ">") >>> > (withId $ ((prepend "!") >>> (append "!"))) > xs = ["test", "foobar"] >>= (runKleisli xform) > mapM_ putStrLn xs

An important observation here is that

f >>> g

is multi-valued composition (g . f), and

(withId f) >>> (withId g) = (returnA <+> f) >>> (returnA <+> g) = ((arr id) <+> f) >>> ((arr id) <+> g)

which, when applied to an input x, returns all values:

((id . id) x) ++ ((id . f) x) ++ ((id . g) x) ++ ((g . f) x) = x ++ (f x) ++ (g x) ++ ((g . f) x)

which are all permutations of using arrows f and g.

## [edit] 7 Tutorial Meta

The wiki file source is literate Haskell. Save the source in a file called ArrowFun.lhs to compile it (or run in GHCi).

The code is adapted to GHC 6.10.1; use [1] for older versions of GHC and other Haskell implementations.

- Original version - Nov 19, 2006, Tim Newsham.

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