I wish I could define something like this:

class (Functor f) => Fixpoint f x | x -> f where

fix :: x -> Fix f

instance (Functor f) => Fixpoint f (forall a. f a) where
fix = id

instance (Functor f, Fixpoint f x) => Fixpoint f (f x) where
fix = Fix . fmap fix

but instances with polymorphic types aren't allowed. (Why is that?)

Alternatively if I could write a function that could turn

e :: forall a. F (F (F ... (F a) ... ))

into

specialize e :: F (F (F ... (F X) ... ))

that would work too, but I don't see how that's possible.

On Mon, May 7, 2012 at 6:59 PM, wren ng thornton <wren@freegeek.org> wrote:

On 5/7/12 8:55 PM, Sebastien Zany wrote:

To slightly alter the question, is there a way to define a class

class (Functor f) => Fixpoint f x where
...

You can just do that (with MPTCs enabled). Though the usability will be much better if you use fundeps or associated types in order to constrain the relation between fs and xs. E.g.:

-- All the following require the laws:
-- > fix . unfix == id
-- > unfix . fix == id

-- With MPTCs and fundeps:
class (Functor f) => Fixpoint f x | f -> x where
fix :: f x -> x
unfix :: x -> f x

class (Functor f) => Fixpoint f x | x -> f where
fix :: f x -> x
unfix :: x -> f x

class (Functor f) => Fixpoint f x | f -> x, x -> f where
fix :: f x -> x
unfix :: x -> f x

-- With associated types:
-- (Add a type/data family if you want both Fix and Pre.)
class (Functor f) => Fixpoint f where
type Fix f :: *
fix :: f (Fix f) -> Fix f
unfix :: Fix f -> f (Fix f)

class (Functor f) => Fixpoint f where
data Fix f :: *
fix :: f (Fix f) -> Fix f
unfix :: Fix f -> f (Fix f)

class (Functor (Pre x)) => Fixpoint x where
type Pre x :: * -> *
fix :: Pre x x -> x
unfix :: x -> Pre x x

class (Functor (Pre x)) => Fixpoint x where
data Pre x :: * -> *
fix :: Pre x x -> x
unfix :: x -> Pre x x

Indeed, that last one is already provided in the fixpoint[1] package, though the names are slightly different. The differences between using "x -> f", "f -> x", or both, and between using "data" or "type", are how it affects inference. That is, implicitly there are two relations on types:

Fix \subseteq * \cross *
Pre \subseteq * \cross *

And we need to know: (1) are these relations functional or not? And, (2) are they injective or not? The answers to those questions will affect how you can infer one of the types (f or x) given the other (x or f).

[1] http://hackage.haskell.org/package/fixpoint

--
Live well,
~wren

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