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Constructive forcing, CPS translations and witness extraction in Interactive realizability†

Published online by Cambridge University Press:  29 October 2015

FEDERICO ASCHIERI
FEDERICO ASCHIERI*
Affiliation:
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Haupstrasse 8-10 1040 Vienna, Austria Email: [email protected]
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Abstract

In Interactive realizability for second-order Heyting Arithmetic with EM1 and SK1 (the excluded middle and Skolem axioms restricted to Σ10-formulas), realizers are written in a classical version of Girard's System F. Since the usual reducibility semantics does not apply to such a system, we introduce a constructive forcing/reducibility semantics: though realizers are not computable functionals in the sense of Girard, they can be forced to be computable. We apply this semantics to show how to extract witnesses for realizable Π20-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/reducibility semantics.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Issue 6 , September 2017 , pp. 993 - 1031
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

†

This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ‘Investissements d'Avenir’ (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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