How can I push polymorphism into a data structure?

Question

A recent question led me to wonder how to convert a

forall f . Functor f => [LensLike f s t a b]

into a

[ReifiedLens s t a b]

There's an easy way to do it really slowly, by indexing into the list with !!, but it's quite incredibly inefficient. It feels like there should be enough parametricity to pull a trick similar to the one used in reflection, but I can't seem to figure anything out. Is it possible to do this efficiently at all?

I don't think this is even possible. We'd have to go (operationally) from Functor f -> [LensLike f s t a b] to [Functor f -> LensLike f s t a b]. We need to pass in a Functor f to pull out a list to begin with, and there isn't one around. — András Kovács
– András Kovács, Commented Oct 12, 2015 at 21:33
@AndrásKovács Perhaps we can use a parametricity argument to convince ourselves that functions of type Functor f -> [LensLike f s t a b] must choose the length of their result list without inspecting the Functor f argument in a meaningful way. Then it's easy to get the rest of the way from there. (Of course this claim is clearly not true of functions of type Functor F -> [LensLike F s t a b] for some particular F.) — Daniel Wagner
– Daniel Wagner, Commented Oct 12, 2015 at 21:35
@AndrásKovacs, that was my fear. We'd need some kind of "superfunctor" dictionary we could pass in that would magically be able to take on the identity of any other Functor dictionary. — dfeuer
– dfeuer, Commented Oct 12, 2015 at 21:36
I'd like to point out for the specific case of lenses, you can use ALens instead of ReifiedLens. I'm wondering if there's a similar trick that works generally... — Ørjan Johansen
– Ørjan Johansen, Commented Oct 12, 2015 at 22:29

Ørjan Johansen · Accepted Answer · 2015-10-12 22:57:13Z

5

Testing out my idea in the comments, you can in fact write this directly by passing through ALens:

convert :: (forall f. Functor f => [LensLike f s t a b]) -> [ReifiedLens s t a b]
convert ls = [Lens (cloneLens l) | l <- ls]

ALens is based on the Pretext functor:

type ALens s t a b = LensLike (Pretext (->) a b) s t a b

newtype Pretext p a b t = Pretext { runPretext :: forall f. Functor f => p a (f b) -> f t }

This allows using a single Functor to represent all of them, at least for lens usage.

I'm still wondering whether there's a method that works generally and not just for lenses.

edited Oct 12, 2015 at 22:57

answered Oct 12, 2015 at 22:51

Ørjan Johansen

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1 Comment

dfeuer Over a year ago

I still find Pretext baffling. As applied to (->), I can sort of see it as a partially applied lens, but its utility and generality still confuse me.

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