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Classical realizability and arithmetical formulæ

Published online by Cambridge University Press:  18 January 2016

MAURICIO GUILLERMO  and
ÉTIENNE MIQUEY
MAURICIO GUILLERMO
Affiliation:
Universidad de la República, IMERL, Facultad de Ingeniería, Montevideo, Uruguay Email: mguille@fing.edu.uy
ÉTIENNE MIQUEY
Affiliation:
Laboratoire PPS, Univ. Paris Diderot, Équipe PiR2, INRIA Universidad de la República, IMERL, Facultad de Ingeniería, Montevideo, Uruguay Email: etienne.miquey@pps.univ-paris-diderot.fr
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Abstract

In this paper, we treat the specification problem in Krivine classical realizability (Krivine 2009 Panoramas et synthèses 27), in the case of arithmetical formulæ. In the continuity of previous works from Miquel and the first author (Guillermo 2008 Jeux de réalisabilité en arithmétique classique, Ph.D. thesis, Université Paris 7; Guillermo and Miquel 2014 Mathematical Structures in Computer Science, Epub ahead of print), we characterize the universal realizers of a formula as being the winning strategies for a game (defined according to the formula). In the first sections, we recall the definition of classical realizability, as well as a few technical results. In Section 5, we introduce in more details the specification problem and the intuition of the game-theoretic point of view we adopt later. We first present a game 1, that we prove to be adequate and complete if the language contains no instructions ‘quote’ (Krivine 2003 Theoretical Computer Science 308 259–276), using interaction constants to do substitution over execution threads. We then show that as soon as the language contain ‘Quote,’ the game is no more complete, and present a second game 2 that is both adequate and complete in the general case. In the last Section, we draw attention to a model-theoretic point of view and use our specification result to show that arithmetical formulæ are absolute for realizability models.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Issue 6 , September 2017 , pp. 1068 - 1107
Copyright
Copyright © Cambridge University Press 2016 

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