A Realizability Interpretation for Intersection and Union Types

Abstract

Proof-functional logical connectives allow reasoning about the structure of logical proofs, in this way giving to the latter the status of first-class objects. This is in contrast to classical truth-functional connectives where the meaning of a compound formula is dependent only on the truth value of its subformulas. In this paper we present a typed lambda calculus, enriched with strong products, strong sums, and a related proof-functional logic. This calculus, directly derived from a typed calculus previously defined by two of the current authors, has been proved isomorphic to the well-known Barbanera-Dezani-Ciancaglini-de'Liguoro type assignment system. We present a logic L ∩∪ featuring two proof-functional connectives, namely strong conjunction and strong disjunction. We prove the typed calculus to be isomorphic to the logic L ∩∪ and we give a realizability semantics using Mints' realizers [Min89] and a completeness theorem. A prototype implementation is also described.

Degrees of Relatedness

“…Intersection and Union Types A line of work for which we can cite [11,21] treats intersection (S ∩ T ) and union (S ∪ T ) types in a manner similar to the product (S ×T ) and the disjoint union (S ⊎T ) respectively, in order to achieve type inference in the presence of intersection and union types. Concretely, the type system is equipped with an erasure function which erases annotations required only for type inference, and A ∩ B is inhabited by pairs (s, t ) for which s : S and t : T have equal erasure; the pair then also erases to the same expression.…”