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The Determinacy of Context-Free Games

By Olivier Finkel

Abstract

This paper is an extended version of a STACS 2012 conference paper. It will appear in the Journal of Symbolic Logic.International audienceWe prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 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 omega-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 A and a Büchi automaton B such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game W(L(A), L(B)) 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(A), L(B))

Topics: models of set theory, independence from the axiomatic system ZFC, context-free games, 1-counter automaton, determinacy, Gale-Stewart games, Wadge games, Automata and formal languages, logic in computer science, [ INFO.INFO-LO ] Computer Science [cs]/Logic in Computer Science [cs.LO], [ INFO.INFO-GT ] Computer Science [cs]/Computer Science and Game Theory [cs.GT], [ MATH.MATH-LO ] Mathematics [math]/Logic [math.LO]
Publisher: Association for Symbolic Logic
Year: 2013
OAI identifier: oai:HAL:hal-00916865v1
Provided by: Hal-Diderot

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