[Haskell-cafe] Restrictions on associated types for classes

[Dan Weston]

I think the denotational meanings are different. The instance also implies:

For each Cl t there must exist a Cl u where u does not unify with [v] 
for some v.
In other words, there must be a ground instance.

For the class declaration, the existence of a ground instance can be 
inferred only by excluding infinite types with strict type unification 
semantics. If infinite types were admitted (where type unification is 
done non-strictly), the class declaration allows for infinite types (let 
t ~ [t] in t). The instance does not.

Dan

Martin Sulzmann wrote:
> The statements
> 
> class Cl [a] => Cl a
> 
> and
> 
> instance Cl a => Cl [a]
> 
> (I omit the type family constructor in the head for simplicyt)
> 
> state the same (logical) property:
> 
> For each Cl t there must exist Cl [t].
> 
> Their operational meaning is different under the dictionary-passing 
> translation [1].
> The instance declaration says we build dictionary Cl [a] given the
> dictionary Cl [a]. The super class declaration says that the dictionary 
> for Cl [a]
> must be derivable (extractable) from Cl a's dictionary. So, here
> we run into a cycle (on the level of terms as well as type inference).
> 
> However, if we'd adopt a type-passing translation [2] (similar to
> dynamic method lookup in oo languages) then there isn't
> necessarily a cycle (for terms and type inference). Of course,
> we still have to verify the 'cyclic' property which smells like
> we run into non-termination if we assume some inductive reason
> (but we might be fine applying co-induction).
> 
> -Martin
> 
> [1] Cordelia V. Hall, Kevin Hammond 
> <http://www.informatik.uni-trier.de/%7Eley/db/indices/a-tree/h/Hammond:Kevin.html>, 
> Simon L. Peyton Jones 
> <http://www.informatik.uni-trier.de/%7Eley/db/indices/a-tree/j/Jones:Simon_L=_Peyton.html>, 
> Philip Wadler 
> <http://www.informatik.uni-trier.de/%7Eley/db/indices/a-tree/w/Wadler:Philip.html>: 
> Type Classes in Haskell. ACM Trans. Program. Lang. Syst. 18 
> <http://www.informatik.uni-trier.de/%7Eley/db/journals/toplas/toplas18.html#HallHJW96>(2): 
> 109-138 (1996)
> 
> [2] Satish R. Thatte: Semantics of Type Classes Revisited. LISP and 
> Functional Programming 1994 
> <http://www.informatik.uni-trier.de/%7Eley/db/conf/lfp/lfp1994.html#Thatte94>: 
> 208-219
> 
> On Thu, Dec 17, 2009 at 6:40 PM, Simon Peyton-Jones 
> <simonpj at microsoft.com <mailto:simonpj at microsoft.com>> wrote:
> 
>     | > Hmm.  If you have
>     | >   class (Diff (D f)) => Diff f where
>     | >
>     | > then if I have
>     | >     f :: Diff f => ...
>     | >     f = e
>     | > then the constraints available for discharging constraints arising
>     | > from e are
>     | >     Diff f
>     | >     Diff (D f)
>     | >     Diff (D (D f))
>     | >     Diff (D (D (D f)))
>     | >     ...
>     | >
>     | > That's a lot of constraints.
>     |
>     | But isn't it a bit like having an instance
>     |
>     |    Diff f => Diff (D f)
> 
>     A little bit.  And indeed, could you not provide such instances?
>      That is, every time you write an equation for D, such as
>            type D (K a) = K Void
>     make sure that Diff (K Void) also holds.
> 
>     The way you it, when you call f :: Diff f => <blah>, you are obliged
>     to pass runtime evidence that (Diff f) holds.  And that runtime
>     evidence includes as a sub-component runtime evidence that (Diff (D
>     f)) holds.   If you like the, the evidence for Diff f looks like this:
>            data Diff f = MkDiff (Diff (D f)) (D f x -> x -> f x)
>     So you are going to have to build an infinite data structure.  You
>     can do that fine in Haskell, but type inference looks jolly hard.
> 
>     For example, suppose we are seeking evidence for
>            Diff (K ())
>     We might get such evidence from either
>      a) using the instance decl
>             instance Diff (K a) where ...
>     or
>      b) using the fact that (D I) ~ K (), we need Diff I, so
>            we could use the instance
>              instance Diff I
> 
>     Having two ways to get the evidence seems quite dodgy to me, even
>     apart from the fact that I have no clue how to do type inference for it.
> 
>     Simon
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