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

Abstract

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 normalizing 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 (B\"ohm trees).

Linear Dependent Types and Relative Completeness

“…In intersection type disciplines, the class of strongly and weakly normalizable lambda terms can be captured [16]. Recently, these results have been refined in such a way that the actual complexity of reduction of the underlying term can be read from its type derivation [14,7].…”

Uniform Proofs of Normalisation and Approximation for Intersection Types

“…By an instance of the (R →) rule, we obtain Thus our method has been successfully applied to proving the approximation theorem for the mapping α and the system λ ∩ω . It is work in progress to give similar proofs of the approximation theorems for the η-approximation mapping α η , which maps λ x.⊥ directly to ⊥, and type systems with various preorders as discussed in [10,11,4].…”

Linear Dependent Types and Relative Completeness

“…The first example that comes to mind is the one of intersection types. In intersection type disciplines, the class of strongly and weakly normalizable lambda terms can be captured [16]. Recently, these results have been refined in such a way that the actual complexity of reduction of the underlying term can be read from its type derivation [14,7].…”