Rebase.Data.Functor.Contravariant.Divisible

A Divisible contravariant functor is the contravariant analogue of Applicative.

Continuing the intuition that Contravariant functors consume input, a Divisible contravariant functor also has the ability to be composed "beside" another contravariant functor.

Serializers provide a good example of Divisible contravariant functors. To begin let's start with the type of serializers for specific types:

newtype Serializer a = Serializer { runSerializer :: a -> ByteString }

This is a contravariant functor:

instance Contravariant Serializer where
  contramap f s = Serializer (runSerializer s . f)

That is, given a serializer for a (s :: Serializer a), and a way to turn bs into as (a mapping f :: b -> a), we have a serializer for b: contramap f s :: Serializer b.

Divisible gives us a way to combine two serializers that focus on different parts of a structure. If we postulate the existance of two primitive serializers - string :: Serializer String and int :: Serializer Int, we would like to be able to combine these into a serializer for pairs of Strings and Ints. How can we do this? Simply run both serializers and combine their output!

data StringAndInt = StringAndInt String Int

stringAndInt :: Serializer StringAndInt
stringAndInt = Serializer $ \(StringAndInt s i) ->
  let sBytes = runSerializer string s
      iBytes = runSerializer int i
  in sBytes <> iBytes

divide is a generalization by also taking a contramap like function to split any a into a pair. This conveniently allows you to target fields of a record, for instance, by extracting the values under two fields and combining them into a tuple.

To complete the example, here is how to write stringAndInt using a Divisible instance:

instance Divisible Serializer where
  conquer = Serializer (const mempty)

  divide toBC bSerializer cSerializer = Serializer $ \a ->
    case toBC a of
      (b, c) ->
        let bBytes = runSerializer bSerializer b
            cBytes = runSerializer cSerializer c
        in bBytes <> cBytes

stringAndInt :: Serializer StringAndInt
stringAndInt =
  divide (\(StringAndInt s i) -> (s, i)) string int

A Decidable contravariant functor is the contravariant analogue of Alternative.

Noting the superclass constraint that f must also be Divisible, a Decidable functor has the ability to "fan out" input, under the intuition that contravariant functors consume input.

In the discussion for Divisible, an example was demonstrated with Serializers, that turn as into ByteStrings. Divisible allowed us to serialize the product of multiple values by concatenation. By making our Serializer also Decidable- we now have the ability to serialize the sum of multiple values - for example different constructors in an ADT.

Consider serializing arbitrary identifiers that can be either Strings or Ints:

data Identifier = StringId String | IntId Int

We know we have serializers for Strings and Ints, but how do we combine them into a Serializer for Identifier? Essentially, our Serializer needs to scrutinise the incoming value and choose how to serialize it:

identifier :: Serializer Identifier
identifier = Serializer $ \identifier ->
  case identifier of
    StringId s -> runSerializer string s
    IntId i -> runSerializer int i

It is exactly this notion of choice that Decidable encodes. Hence if we add an instance of Decidable for Serializer...

instance Decidable Serializer where
  lose f = Serializer $ \a -> absurd (f a)
  choose split l r = Serializer $ \a ->
    either (runSerializer l) (runSerializer r) (split a)

Then our identifier Serializer is

identifier :: Serializer Identifier
identifier = choose toEither string int where
  toEither (StringId s) = Left s
  toEither (IntId i) = Right i

Redundant, but provided for symmetry.

conquered = conquer

This is the divisible analogue of liftA. It gives a viable default definition for contramap in terms of the members of Divisible.

liftD f = divide ((,) () . f) conquer