Thanks for the thorough answer, Dan. That's exactly what I was looking for. During further search, I stumbled on an excellent introductory description of recursive types in a draft of Robert Harper's book "Programming Languages: Theory and Practice" http://www.cs.cmu.edu/~rwh/plbook/book.pdf -- vi On Fri, Dec 25, 2009 at 10:48 PM, Dan Doel <dan.doel@gmail.com> wrote:

On Friday 25 December 2009 11:35:38 am Vladimir Ivanov wrote:

...

Dear All,

Recently, I've been playing with self-application and fixed-point combinators definition in Haskell.

It is possible to type them in Haskell using recursive types.

I took Y & U combinators:

...

newtype Rec a = In { out :: Rec a -> a }

u :: Rec a -> a u x = out x x

y :: (a -> a) -> a y f = let uf = f . u in uf (In uf)

Recursive types are part of System F-omega, which corresponds to lambda omega in Barendregt's Lambda Cube. For all type systems in Lambda Cube it is proven that they are strongly normalizing. But using "y" I can easily construct a term even without a normal form.

So, I got a contradition and conclude that type system implementation in Haskell contains something, that is absent from System F-omega.

My question is: what's particular feature in Haskell type system, that breaks strong normalization property? Or am I totally wrong classifying it in terms of Lambda Cube?

General recursive types are not part of F_omega. It only has...

   1) Type functions of arbitrary order      /\ F : * -> *. /\ T : *. F T    2) Universal quantification over the above spaces      Pi F : * -> *. Pi T : (* -> *) -> *. T F

You can encode inductive and coinductive types in such a type system. If F is the shape functor, then the encoding of the inductive type is:

 Pi R : *. (F R -> R) -> R

and the encoding of the coinductive type is:

 Ex R : *. F R * R

where:

 Ex R : *. T[R] = Pi Q : *. (Pi R : *. T[R] -> Q) -> Q

and:

  A * B = Pi R : *. (A -> B -> R) -> R

(and

  A + B = Pi R : *. (A -> R) -> (B -> R) -> R

if you need it to encode F R, although often one may be able to massage F into a more suitable form to avoid using the * and + encodings).

However, as you can imagine, Rec cannot be encoded. Allowing such types adds general recursion back in. In type theories that do support Haskell-alike algebraic/(co)inductive definitions (which none in the lambda cube do; they're plain lambda calculi), the total ones restrict what types you can declare (to strictly positive types, usually) to make types like Rec illegal.

-- Dan