Typing the Y combinator

Question

http://muaddibspace.blogspot.com/2008/01/type-inference-for-simply-typed-lambda.html is a concise definition of the simply typed lambda calculus in Prolog.

It looks okay, but then he purports to assign a type to the Y combinator... whereas in a very real sense the entire purpose of adding types to lambda calculus is to refuse to assign a type to things like the Y combinator.

Can anyone see exactly where his error or -- more likely -- my misunderstanding is?

Answer 1

The Y combinator in its basic form

Y f = (\x -> f (x x)) (\x -> f (x x))

just cannot be typed using the simple type system proposed in the article.

There are other, much easier but meaningful examples that cannot be typed on that level:

Take e.g.

test f = (f 1, f "Hello")

This obviously works for test (\x -> x) but we cannot give the higher-ranked type that was required here, namely

test :: (∀a . a -> a) -> (Int, String)  

But even in more advanced type systems like the GHCI extensions of Haskell which allow the above, Y is still hard to type.

So, given the possibility of recursion, we can just define and work using the fix combinator

fix f = f (fix f) 

with fix :: (a -> a) -> a