2013-12-22-24-days-of-hackage-profunctors.md

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24 Days of Hackage: profunctors

Yesterday, Tom showed us a different type of functor than the ordinary Haskell Functor - the contravariant functor. Today, Tom's going to guide us through another type of functor - the profunctor.

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Yesterday, we considered the intuition that functors are producers of output, and contravariant functors are consumers of input - and both functors can be adapted to work with different types. What about a datatype that represents both an "adaptable consumer of input" and an "adaptable producer of output" at the same time, i.e. some sort of "pipe" structure? This is exactly what a Profunctor instance is, and again the function arrow a -> b gives us our prototypical example of such a type. A Profunctor has two type parameters, a and b. The first can be thought of as an "input of type a" and can be adapted with a function of type c -> a, like Contravariant. The second can be thought of as an "output of type b" and can be adapted with a function of type b -> d, like Functor. This gives us the following type class:

class Profunctor p where
  lmap :: (c -> a) -> p a b -> p c b
  rmap :: (b -> d) -> p a b -> p a d
  dimap :: (c -> a) -> (b -> d) -> p a b -> p c d

lmap adapts the input (left hand type variable) and rmap adapts the output (right hand type variable). dimap adapts them both at the same time.

Much like Functor and Contravariant, Profunctor instances must satisfy some laws. The laws that a Profunctor must satisfy are a combination of the Functor and Contravariant laws for its type parameters:

• dimap id id = id
• dimap (h' . h) (f . f') = dimap h f . dimap h' f'

Furthermore, rmap f should be equivalent to dimap id f, and lmap f should be equivalent to dimap f id. Because of this, the minimal definition of a Profunctor instance is either specifying rmap and lmap in terms of dimap, or dimap in terms of rmap and lmap.

For functions, the Profunctor instance is:

instance Profunctor (->) where
  lmap h g = g . h
  rmap f g = f . g
  dimap h f g = f . g . h

If you study this, you'll see that lmap adapts a function g by composing h on the right (changing the "input"), while rmap adapts g by composing f on the left (changing the "output"). Using equational reasoning, we can easily prove that this instance does indeed satisfy the laws:

dimap id id = \g -> id . g . id = \g -> g = id
dimap (h' . h) (f . f') = \g -> (f . f') . g . (h' . h)
                        = \g -> f . (f' . g . h') . h
                        = \g -> f . dimap h' f' g . h
                        = \g -> dimap h f (dimap h' f' g)
                        = dimap h f . dimap h' f'

So all is well.

The function arrow is the prototypical example of a profunctor, but it is also probably the most boring one. Here's a more interesting example: a datatype that represents the concept of a left fold.

data L a b = forall s. L (s -> b) (s -> a -> s) s

The left fold is a process that updates state according to its input, and can transform this state into a final result. The datatype contains some value of type s representing the initial state, a map of type s -> a -> s which reads a value a and updates the state accordingly, and a map of type s -> b which converts the final state into the return value. This is a Profunctor or "adaptable pipe with input of type a and output of type b". (You'll find this definition in the folds package).

instance Profunctor L where
  dimap h f (L result iterate initial) =
    L (f . result) (\s -> iterate s . h) initial

runFold :: L a b -> [a] -> b
runFold (L result iterate initial) = result . foldl iterate initial

Here's a left fold that represents a sum.

summer :: Num a => L a a
summer = L id (+) 0

testSummer :: Int
testSummer = runFold summer [1..10]

> testSummer
55

This is fine if we're summing a list of Num instances, but can we use this to fold other types of lists? Of course we can, and to do so we'll need to change the input type to our fold. Here's an example of adapting the left fold to a different input and output type.

lengther :: L String String
lengther = dimap length (\s -> "The total length was " ++ show s) summer

testLengther :: String
testLengther = runFold lengther ["24", "days", "of", "hackage", "!"]

> testLengther
"The total length was 16"

There are not many Profunctor definitions on hackage, although they are used in the internals of lens. Personally I have used them heavily in Opaleye, a relation database EDSL. In Opaleye there are profunctors which act exactly as the intuition about them suggests: they can be seen as consuming values of one type and producing values of another. For example there is a Profunctor instance for "consuming" the rows returned by a query running on PostgreSQL and "producing" the equivalent values in Haskell.

Defining a Contravariant or Profunctor instance for your datatype can give you more certainty about the correctness of your code, just like defining a Functor instance can. Unless I am much mistaken, parametricity implies that there is at most one valid Contravariant or Profunctor instance for your datatype. Thus defining such an instance cannot restrict you in any way, and acts as an additional property for the type checker to check that your program satisfies. If you expected your type to be contravariant in its argument, or a profunctor in two arguments, but you can't write a definition to satisfy the compiler then perhaps you have a bug in your type definition!

Have a look through your code and if you find suitable datatypes, give yourself the gift of Contravariant and Profunctor instances this Christmas.

Thanks to merijn on the #haskell IRC channel who suggested the output/input/pipe adaptor analogy.