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A direct computational interpretation of second-order arithmetic via update recursion

Abstract

Second-order arithmetic has two kinds of computational interpretations: via Spector's bar recursion of via Girard's polymorphic lambda-calculus. Bar recursion interprets the negative translation of the axiom of choice which, combined with an interpretation of the negative translation of the excluded middle, gives a computational interpretation of the negative translation of the axiom scheme of comprehension. It is then possible to instantiate universally quantified sets with arbitrary formulas (second-order elimination). On the other hand, polymorphic lambda-calculus interprets directly second-order elimination by means of polymorphic types. The present work aims at bridging the gap between these two interpretations by interpreting directly second-order elimination through update recursion, which is a variant of bar recursion.
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Dates and versions

hal-03698879 , version 1 (19-06-2022)

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Valentin Blot. A direct computational interpretation of second-order arithmetic via update recursion. LICS 2022 - 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Aug 2022, Haïfa, Israel. ⟨10.1145/3531130.3532458⟩. ⟨hal-03698879⟩
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