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Interactive Realizability for Second-Order Heyting Arithmetic with EM1 and SK1

By Federico Aschieri

Abstract

We introduce a classical realizability semantics based on interactive learning for full second-order Heyting Arithmetic with excluded middle and Skolem axioms over Sigma01-formulas. Realizers are written in a classical version of Girard's System F. Since the usual computability semantics does not apply to such a system, we introduce a constructive forcing/computability semantics: though realizers are not computable functional in the sense of Girard, they can be forced to be computable. We apply these semantics to show how to extract witnesses from realizable Pi02-formulas. In particular a constructive and efficient method is introduced. It is based on a new ''(state-extending-continuation)-passing-style translation'' whose properties are described with the constructive forcing/computability semantics

Topics: ACM : F.: Theory of Computation, ACM : F.: Theory of Computation/F.3: LOGICS AND MEANINGS OF PROGRAMS, ACM : F.: Theory of Computation/F.4: MATHEMATICAL LOGIC AND FORMAL LANGUAGES/F.4.1: Mathematical Logic/F.4.1.7: Proof theory, ACM : F.: Theory of Computation/F.3: LOGICS AND MEANINGS OF PROGRAMS/F.3.1: Specifying and Verifying and Reasoning about Programs/F.3.1.2: Logics of programs, [ INFO.INFO-LO ] Computer Science [cs]/Logic in Computer Science [cs.LO]
Publisher: HAL CCSD
Year: 2012
OAI identifier: oai:HAL:hal-00657054v2
Provided by: Hal-Diderot

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