Infinitary intersection types as sequences: A new answer to Klop's problem

Abstract

We provide a type-theoretical characterization of weakly-normalizing terms in an infinitary lambda-calculus. We adapt for this purpose the standard quantitative (with non-idempotent intersections) type assignment system of the lambda-calculus to our infinite calculus.Our work provides a positive answer to a semi-open question known as Klop's Problem, namely, finding out if there is a type system characterizing the set of hereditary head-normalizing (HHN) lambda-terms. Tatsuta showed in 2007 that HHN could not be characterized by a finite type system. We prove that an infinitary type system endowed with a validity condition called approximability can achieve it.As it turns out, approximability cannot be expressed when intersection is represented by means of multisets. Multisets are then replaced coinductively by sequences of types indexed by integers, thus defining a type system called System S.