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A Computational Interpretation of the Axiom of Determinacy in Arithmetic

Authors: Takanori Hida

Published in: LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)


Abstract
We investigate the computational content of the axiom of determinacy (AD) in the setting of classical arithmetic in all finite types with the principle of dependent choices (DC). By employing the notion of realizability interpretation for arithmetic given by Berardi, Bezem and Coquand (1998), we interpret the negative translation of AD. Consequently, the combination of the negative translation with this realizability semantics can be seen as a model of DC, AD and the negation of the axiom of choice at higher types. In order to understand the computational content of AD, we explain, employing Coquand's game theoretical semantics, how our realizer behaves.

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Takanori Hida. A Computational Interpretation of the Axiom of Determinacy in Arithmetic. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 335-349, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)


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@InProceedings{hida:LIPIcs.CSL.2012.335,
  author =	{Hida, Takanori},
  title =	{{A Computational Interpretation of the Axiom of Determinacy in Arithmetic}},
  booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
  pages =	{335--349},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-42-2},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{16},
  editor =	{C\'{e}gielski, Patrick and Durand, Arnaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.335},
  URN =		{urn:nbn:de:0030-drops-36828},
  doi =		{10.4230/LIPIcs.CSL.2012.335},
  annote =	{Keywords: The axiom of determinacy, Gale-Stewart’s theorem, Syntactic continuity, Realizability interpretation, Coquand’s game semantics}
}
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