Arrow tutorial

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

The Arrow class

A value of type (Arrow a) => a b c (commonly referred to as just an arrow) represents a process that takes as input a value of type b and outputs a value of type c.

The class includes the following methods:

• arr builds an arrow value out of a function:

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

• (>>>) composes two arrow values to form a new one by "chaining" them together, one after another:

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

• first and second make a new arrow value out of an existing one. They perform a transformation (given by their argument) on either the first or the second item of a pair:

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.

A simple arrow type

Let's define a really simple arrow type as an example, based on a function mapping an input to an output:

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

Some other arrow operations

Now let's define some operations that are generic to all arrow types:

• split is an arrow value 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 value 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       
     -- = \op -> arr (\(x,y) -> x `op` y)

• (***) combines two arrow values by running them on a pair (the first arrow value on the first component of the pair; the second arrow value on the second component of the pair):

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

• (&&&) combines two arrow values by running them with the same input:

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

• liftA2 makes a new arrow value that combines the output from two other arrow values using a binary operation. It works by splitting a value and operating on both halves 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

An example

Now let's build something using our simple arrow definition and some of the tools we've just created. We start with two simple arrow values, f and g:

• f halves its input:

f :: SimpleFunc Int Int
f = arr (`div` 2)

• and g triples its input and adds one:

g :: SimpleFunc Int Int
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 (+). Let's work through an application of h to the value 8:

8	→	(8, 8)	split
(8, 8)	→	(4, 8)	first f ⇔ x `div` 2, where x is the first component of the pair
(4, 8)	→	(4, 25)	second g ⇔ 3*y + 1, where y is the second component of the pair
(4, 25)	→	29	apply (+) to the components of the pair

f ↗ ↘ 8 → (split) (unsplit (+)) → 29 ↘ ↗ g

We can see that h is a new arrow value that, when applied to 8, will apply both f and g to 8, then adds their results.

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

Kleisli arrow values

Let's move on to something a little fancier now: Kleisli arrows.

A Kleisli arrow type (Kleisli m a b) corresponds to the type (a -> m b), where m is a monadic type. 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, and second. This means that all multi-value functions (i.e of type a -> [b]) are already defined as Kleisli arrows (because the list type [] is monadic)! (>>>) performs composition, keeping track of all the multiple results. split, (&&&) and (***) are all defined as before. 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

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 type is an arrow value that always returns the empty list) and a (<+>) operator (which takes the results from two arrow values and concatenates them). We can build up some pretty interesting string transformations (multi-valued functions of type String -> [String]) using Kleisli arrow values:

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 a 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 the arrow values f and g.

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.