Relating Typability and Expressiveness in Finite-Rank Intersection Type Systems

We investigate finite-rank intersection type systems, analyzing the complexity of their type inference problems and their relation to the problem of recognizing semantically equivalent terms. Intersection types allow something of type τ1 ∧ τ2 to be used in some places at type τ1 and in other places at type τ2. A finite-rank intersection type system bounds how deeply the ∧ can appear in type expressions. Such type systems enjoy strong normalization, subject reduction, and computable type inference, and they support a pragmatics for implementing parametric polymorphism. As a consequence, they provide a conceptually simple and tractable alternative to the impredicative polymorphism of System F and its extensions, while typing many more programs than the Hindley-Milner type system found in ML and Haskell. While type inference is computable at every rank, we show that its complexity grows exponentially as rank increases. Let K(0, n) = n and K(t + 1, n) = 2; we prove that recognizing the pure λ-terms of size n that are typable at rank k is complete for dtime[K(k−1, n)]. We then consider the problem of deciding whether two λ-terms typable at rank k have the same normal form, Supported by NATO grant CRG 971607, NSF grant CCR– 9417382, and NSF grant EIA–9806747. Supported by ONR Grant N00014-93-1-1015, NSF Grant CCR-9619638, and the Tyson Foundation. Supported by NSF grant EIA–9806747. Supported by EPSRC grant GR/L 36963, and NSF grant