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A Realizability Interpretation for Intersection and Union Types

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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 con- nectives 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 cal- culus, 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.
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Dates and versions

hal-01317213 , version 1 (18-05-2016)
hal-01317213 , version 2 (09-09-2016)

Identifiers

  • HAL Id : hal-01317213 , version 2

Cite

Daniel J. Dougherty, Ugo de 'Liguoro, Luigi Liquori, Claude Stolze. A Realizability Interpretation for Intersection and Union Types. 14th Asian Symposium on Programming Languages and Systems, Nov 2016, Hanoi, Vietnam. ⟨hal-01317213v2⟩
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