[Haskell-cafe] Aren't type system extensions fun? [Further analysis]

[Edsko de Vries]

On Thu, May 29, 2008 at 01:44:24PM +0200, Roberto Zunino wrote:
> Kim-Ee Yeoh wrote:
> >
> >How about
> >  foo :: (exists. m :: * -> *. forall a. a -> m a) -> (m Char, m Bool)
> 
> Thank you: I had actually thought about something like that.
> 
> First, the exists above should actually span over the whole type, so it 
> becomes a forall because (->) is contravariant on its 1st argument:
> 
>    foo :: forall m :: * -> * . (forall a. a -> m a) -> (m Char, m Bool)
> 
> This seems to be Haskell+extensions, but note that m above is meant to 
> be an arbitrary type-level function, and not a type constructor (in 
> general).  So, I am not surprised if undecidability issues arise in type 
> checking/inference. :-)

Type *checking* is still decidable (System Fw is sufficiently powerful
to model these types) but type inference now needs higher order
unification, which indeed is undecidable.

> Another valid type for foo can be done AFAICS with intersection types:
> 
>    foo :: (Char -> a /\ Bool -> b) -> (a,b)
> 
> But I can not comment about their inference, or usefulness in practice.

Again, undecidable :) In fact, I believe that an inference algorithm for
intersection types is equivalent to solving the halting problem. Type
checking however is decidable, but expensive.

Edsko