INFINITE LAMBDA CALCULUS AND TYPES

Recent work on infinitary versions of the lambda calculus has shown that the infinite lambda calculus can be a useful tool to study the unsolvable terms of the classical lambda calculus. Working in the framework of the intersection type disciplines, we devise a type assignment system such that two terms are equal in the infinite lambda calculus iff they can be assigned the same types in any basis. A novel feature of the system is the presence of a type constant to denote the set of all terms of order zero, and the possibility of applying a type to another type. We prove a completeness and an approximation theorem for our system. Our results can be considered as a first step towards the goal of giving a denotational semantics for the lambda calculus which is suited for the study of the unsolvable terms. However, some noncontinuity phenomena of the infinite lambda calculus make a full realization construction of a filter model) a quite difficult task