Haskell中泛型多态ADT的Functor实例？

Question

When it comes to applying category theory for generic programming Haskell does a very good job, for instance with libraries like recursion-schemes . However one thing I'm not sure of is how to create a generic functor instance for polymorphic types.

If you have a polymorphic type, like a List or a Tree, you can create a functor from (Hask × Hask) to Hask that represents them. For example:

data ListF a b = NilF | ConsF a b  -- L(A,B) = 1+A×B
data TreeF a b = EmptyF | NodeF a b b -- T(A,B) = 1+A×B×B

These types are polymorphic on A but are fixed points regarding B, something like this:

newtype Fix f = Fix { unFix :: f (Fix f) }
type List a = Fix (ListF a)
type Tree a = Fix (TreeF a)

But as most know, lists and trees are also functors in the usual sense, where they represent a "container" of a 's, which you can map a function f :: a -> b to get a container of b 's.

I'm trying to figure out if there's a way to make these types (the fixed points) an instance of Functor in a generic way, but I'm not sure how. I've encountered the following 2 problems so far:

---

1) First, there has to be a way to define a generic gmap over any polymorphic fixed point. Knowing that types such as ListF and TreeF are Bifunctors, so far I've got this:

{-# LANGUAGE ScopedTypeVariables #-}
import Data.Bifunctor

newtype Fix f = Fix { unFix :: f (Fix f) }

cata :: Functor f => (f a -> a) -> Fix f -> a
cata f = f . fmap (cata f) . unFix

-- To explicitly use inF as the initial algebra
inF :: f (Fix f) -> Fix f
inF = Fix

gmap :: forall a b f. Bifunctor f => (a -> b) -> Fix (f a) -> Fix (f b)
gmap f = cata alg
    where
        alg :: f a (Fix (f b)) -> Fix (f b)
        alg = inF . bimap f id

In Haskell this gives me the following error: Could not deduce (Functor (fa)) arising from a use of cata from the context (Bifunctor f) .

I'm using the bifunctors package, which has a WrappedBifunctor type that specifically defines the following instance which could solve the above problem: Bifunctor p => Functor (WrappedBifunctor pa) . However, I'm not sure how to "lift" this type inside Fix to be able to use it

2) Even if the generic gmap above can be defined, I don't know if it's possible to create a generic instance of Functor that has fmap = gmap , and can instantly work for both the List and Tree types up there (as well as any other type defined in a similar fashion). Is this possible?

If so, would it be possible to make this compatible with recursion-schemes too?

Answer 1

If you're willing to accept for the moment you're dealing with bifunctors, you can say

cata :: Bifunctor f => (f a r -> r) -> Fix (f a) -> r
cata f = f . bimap id (cata f) . unFix

and then

gmap :: forall a b f. Bifunctor f => (a -> b) -> Fix (f a) -> Fix (f b)
gmap f = cata alg
    where
        alg :: f a (Fix (f b)) -> Fix (f b)
        alg = inF . bimap f id

(In gmap , I've just rearranged your class constraint to make scoped type variables work.)

You can also work with your original version of cata , but then you need both the Functor and the Bifunctor constraint on gmap :

gmap :: forall a b f. (Bifunctor f, Functor (f a)) => (a -> b) -> Fix (f a) -> Fix (f b)
gmap f = cata alg
    where
        alg :: f a (Fix (f b)) -> Fix (f b)
        alg = inF . bimap f id

You cannot make your gmap an instance of the normal Functor class, because it would need to be something like

instance ... => Functor (\ x -> Fix (f x))

and we don't have type-level lambda. You can do this if you reverse the two arguments of f , but then you lose the "other" Functor instance and need to define cata specific for Bifunctor again.

[You might also be interested to read http://www.andres-loeh.de/IndexedFunctors/ for a more general approach.]

Answer 2

TBH I'm not sure how helpful this solution is to you because it still requires an extra newtype wrapping for these fixed-point functors, but here we go:

You can keep using your generic cata if you do some wrapping/unwrapping

Given the following two helper functions:

unwrapFixBifunctor :: (Bifunctor f) => Fix (WrappedBifunctor f a) -> Fix (f a)
unwrapFixBifunctor = Fix . unwrapBifunctor . fmap unwrapFixBifunctor . unFix

wrapFixBifunctor :: (Bifunctor f) => Fix (f a) -> Fix (WrappedBifunctor f a)
wrapFixBifunctor = Fix . fmap wrapFixBifunctor . WrapBifunctor . unFix

you can define gmap without any additional constraint on f :

gmap :: (Bifunctor f) => (a -> b) -> Fix (f a) -> Fix (f b)
gmap f = unwrapFixBifunctor . cata alg . wrapFixBifunctor
  where
    alg = inF . bimap f id

You can make Fix . f Fix . f into a Functor via a newtype

We can implement a Functor instance for \\a -> Fix (fa) by implementing this "type-level lambda" as a newtype :

newtype FixF f a = FixF{ unFixF :: Fix (f a) }

instance (Bifunctor f) => Functor (FixF f) where
    fmap f = FixF . gmap f . unFixF

Answer 3

The bifunctors package also offers a version of Fix that's especially appropriate:

newtype Fix p a = In {out :: p (Fix p a) a}

This is made a Functor instance rather easily:

instance Bifunctor p => Functor (Fix p) where
  fmap f = In . bimap (fmap f) f . out