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The Determinacy of Context-Free Games

Published online by Cambridge University Press:  12 March 2014

Olivier Finkel
Olivier Finkel*
Affiliation:
Equipe de Logique Mathématique, Institut de Mathématiques de Jussieu - Paris Rive Gauche UMR 7586, CNRS et Université Paris Diderot Paris 7, Bâtiment Sophie Germain Case 7012, 75205 Paris Cedex 13, France, E-mail: finkel@math.univ-paris-diderot.fr
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Abstract

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by realtime 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of ω-languages accepted by 1-counter Büchi automata is equivalent to the (effective) analytic Wadge determinacy. Using some results of set theory we prove that one can effectively construct a 1-counter Büchi automaton and a Büchi automaton such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(), L()); (2) There exists a model of ZFC in which the Wadge game W(L(), L()) is not determined. Moreover these are the only two possibilities, i.e. there are no models of ZFC in which Player 1 has a winning strategy in the Wadge game W(L(), L()).

Keywords

Automata and formal languageslogic in computer scienceGale-Stewart gamesWadge gamesdeterminacyeffective analytic determinacycontext-free games1-counter automatonmodels of set theoryindependence from the axiomatic system ZFC

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1115 - 1134
Copyright
Copyright © Association for Symbolic Logic 2013

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