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

Federico Aschieri 1
1 PI.R2 - Design, study and implementation of languages for proofs and programs
PPS - Preuves, Programmes et Systèmes, Inria Paris-Rocquencourt, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique : UMR7126
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.
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Contributor : Federico Aschieri <>
Submitted on : Monday, March 5, 2012 - 10:02:08 PM
Last modification on : Friday, January 4, 2019 - 5:33:25 PM
Long-term archiving on : Wednesday, June 6, 2012 - 2:32:11 AM


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  • HAL Id : hal-00657054, version 2



Federico Aschieri. Interactive Realizability for Second-Order Heyting Arithmetic with EM1 and SK1. 2012. ⟨hal-00657054v2⟩



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