Intersection Types, $\lambda$-models, and Böhm Trees

Abstract

This paper is an introduction to intersection type disciplines, with the aim of illustrating their theoretical relevance in the foundations of $\lambda$-calculus. We start by describing the well-known results showing the deep connection between intersection type systems and normalization properties, i.e., their power of naturally characterizing solvable, normalizing, and strongly normal- izing pure $\lambda$-terms. We then explain the importance of intersection types for the semantics of $\lambda$-calculus, through the construction of filter models and the representation of algebraic lattices. We end with an original result that shows how intersection types also allow to naturally characterize tree representations of unfoldings of $\lambda$-terms (Böohm trees).