Squinting at fusion

This being my first blog post and all, I’ll try to maximise boredom and minimise readability by writing as few lines of text as possible. Here we go…

As we know, recursive data types are fixpoints of non-recursive ones. So, for instance, the standard list data type:

data [a] = [] | a : [a]

is just the fixpoint of this guy:

data PreList a s = Nil | Cons a s

with these simple injection/projection functions:

inject :: PreList a [a] -> [a]
inject Nil         = []
inject (Cons x xs) = x : xs

project :: [a] -> PreList a [a]
project []     = Nil
project (x:xs) = Cons x xs

Of course, PreList is also a functor:

instance Functor (PreList a) where
  fmap f Nil        = Nil
  fmap f (Cons x s) = Cons x (f s)

Now, our goal is to mutilate the standard short cut fusion rules by using PreList as much as possible. Let’s do destroy/unfoldr first:

destroy :: (forall s. (s -> PreList a s) -> s -> t) -> [a] -> t
destroy g = g project

unfoldr :: (s -> PreList a s) -> s -> [a]
unfoldr f = inject . fmap (unfoldr f) . f

The fusion rule is

destroy g (unfoldr f s) = g f s

Of course, unfoldr is just the list anamorphism but what’s so special about destroy? If we squint hard enough at its type we might realise that

forall t. (forall s. (s -> PreList a s) -> s -> t) -> t

is, in fact, isomorphic to

exists s. (s -> PreList a s, s)

This being Haskell, we have to introduce a separate data type for the existential:

data Unfolding a = forall s. Unfolding (s -> PreList a s) s

This makes the signatures of our two functions a lot simpler:

destroy :: [a] -> Unfolding a
destroy xs = Unfolding project xs

unfoldr :: Unfolding a -> [a]
unfoldr (Unfolding f s) = ana s
  where
     ana = inject . fmap ana . f

And the fusion rule is a bit nicer, too:

destroy (unfoldr s) = s

In fact, what we have here is almost but not quite stream fusion: destroy is equivalent to stream, unfoldr to unstream and Unfolding to Stream. The only difference is that PreList (which corresponds to the Step data type in the paper) is missing the Skip constructor which, unfortunately, is crucial for making the whole thing work.

This part was clear to me back when we were doing stream fusion but the next bit I didn’t understand until recently. The question is: can we do something similar with foldr/build? Again, let’s get rid of as many funny types as possible and use PreList instead:

foldr :: (PreList a s -> s) -> [a] -> s
foldr f = f . fmap (foldr f) . project

build :: (forall s. (PreList a s -> s) -> s) -> [a]
build g = g inject

If you are wondering where these signatures come from, here is a little hint:

(a -> s -> s) -> s -> t   ~   ((a,s) -> s) -> (() -> s) -> t    ~   ((a,s)+() -> s) -> t   ~   (PreList a s -> s) -> t

Now, by squinting at the types we might just see that it makes sense to introduce an abstraction:

data Folding a = Folding (forall s. (PreList a s -> s) -> s)

and use it:

foldr :: [a] -> Folding a
foldr xs = Folding (\f -> let cata = f . fmap cata . project in cata xs)

build :: Folding a -> [a]
build (Folding g) = g inject

Voilà! We now have a shiny new foldr/build fusion rule:

foldr (build s) = s

It’s probably worth pointing out that Folding does, in fact, support many useful list operations. Here is an example:

instance Functor Folding where
  fmap f (Folding g) = Folding (\h -> g (h . emap))
    where
      emap Nil        = Nil
      emap (Cons x s) = Cons (f x) s

So is there a point to all this? Well, I find it quite interesting that foldr/build fusion can be rewritten in this way. I’m even more intrigued by what we get if we squint at the list hylomorphism:

refold :: Unfolding a -> Folding a
refold (Unfolding g s) = Folding (\f -> let hylo = f . fmap hylo . g in  hylo s)

Could we have both stream fusion and foldr/build in one framework? Would that be useful? Is there a way of working on fusion without irreparably damaging my eyes?