the MPTC Dilemma (please solve)
Manuel M T Chakravarty
chak at cse.unsw.edu.au
Sun Mar 19 11:39:55 EST 2006
Jean-Philippe Bernardy:
> On 3/18/06, Manuel M T Chakravarty <chak at cse.unsw.edu.au> wrote:
> > Here addition and multiplication on Peano numerals using MPTCs and FDs:
> >
> > data Z
> > data S a
> >
> > class Add a b c | a b -> c
> > instance Add Z b b
> > instance Add a b c => Add (S a) b (S c)
> >
> > class Mul a b c | a b -> c
> > instance Mul Z b Z
> > instance (Mul a b c, Add c b d) => Mul (S a) b d
> >
> > It's a mess, isn't it. Besides, this is really untyped programming.
> > You can add instances with arguments of types other than Z and S without
> > the compiler complaining. So, we are not simply using type classes for
> > logic programming, we use them for untyped logic programming. Not very
> > Haskell-ish if you ask me.
> >
> > I'd rather write the following:
> >
> > kind Nat = Z | S Nat
> >
> > type Add Z (b :: Nat) = b
> > Add (S a) (b :: Nat) = S (Add a b)
> >
> > type Mul Z (b :: Nat) = Z
> > Mul (S a) (b :: Nat) = Add (Mul a b) b
> >
> > Well, actually, I'd like infix type constructors and some other
> > syntactic improvements, but you get the idea. It's functional and
> > typesafe. Much nicer.
>
> Indeed, this looks all very good and I truly believe this is the
> direction we should move in.
>
> Still, a question: do you propose to go as far as to replace the
> "traditional" single parameter type-classes with /partial/ type
> constructors?
No. Type classes are fine as they are. However, if you want to
associate types with classes, where the definition of these types varies
on an instance by instance basis, I argue that associated types (ie,
type definitions in classes) are more appropriate than MPTCs with
FDs[*]:
http://hackage.haskell.org/trac/haskell-prime/wiki/AssociatedTypes
Associated types are *open* definitions of type functions (ie, you can
always extend them by adding more instances). In some situations
*closed* definitions of type functions are preferable (as in the Peano
arithmetic example above, which can for example be nicely combined with
GADTs to define bounded lists). AFAIK nobody has a worked out a
proposal for how to add closed type functions to Haskell, but Tim Sheard
has argued for their usefulness in his language Omega:
http://www.cs.pdx.edu/~sheard/papers/LangOfTheFuture.ps
Manuel
[*] I am unsure whether we want MPTCs if we have associated types. It
seems that the typical uses of MPTCs become single parameter classes
when associated types are used instead of FDs.
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