Hyperformulae, Parallel Deductions and Intersection Types

Abstract

We aim at investigating the intersection-type assignment system for lambda calculus, with the Curry-Howard approach. We devise a propositional logic, whose notable characteristic is the presence of the hyperformulae denoting parallel compositions of formulae. As such, this logic formalizes a novel notion of parallel deductions, while forming a simple generalization of the standard natural deduction framework.We prove that the logical calculus is isomorphic to the intersection type system, by mapping logical deductions into typed lambda terms, encoding those deductions, and conversely. In this context the intersection type constructor, which comes out to be a proof-theoretic operator, is now interpreted as a standard propositional connective.