Is there a typed SKI calculus?

Question

Most of us know the correspondence between combinatory logic and lambda calculus. But I've never seen (maybe I haven't looked deep enough) the equivalent of "typed combinators", corresponding to the simply typed lambda calculus. Does such thing exist? Where could one find information about it?

Answer 1

The expressive completeness of the typed combinators compared to the simply typed lambda calculus has been demonstrated. For each untyped combinator, one needs a whole family of typed combinators. For example, one has

• $\mathbf{I}_{\alpha\to\alpha}$
• $\mathbf{K}_{\alpha\to(\beta\to\alpha)}$
• $\mathbf{S}_{\alpha\to(\beta\to\gamma)\to(\alpha\to\beta\to(\alpha\to\gamma))}$

for all combinations of simple types $\alpha,\beta$ and $\gamma$.

Alternatively, just think of the types as type schemes (or polymorphic types) and enter them into Haskell and voila: combinators.