Journal of Symbolic Logic

The determinacy of context-free games

Olivier Finkel

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We 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 $\mathcal{A}$ and a Büchi automaton $\mathcal{B}$ such that: (1) There exists a model of ZFC in which Player 2 has a winning strategy in the Wadge game $W(L(\mathcal{A}), L(\mathcal{B}))$; (2) There exists a model of ZFC in which the Wadge game $W(L(\mathcal{A}), L(\mathcal{B}))$ 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 $W(L(\mathcal{A}), L(\mathcal{B}))$.

Article information

J. Symbolic Logic, Volume 78, Issue 4 (2013), 1115-1134.

First available in Project Euclid: 5 January 2014

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Zentralblatt MATH identifier

Automata and formal languages logic in computer science Gale-Stewart games Wadge games determinacy effective analytic determinacy context-free games 1-counter automaton models of set theory independence from the axiomatic system ZFC


Finkel, Olivier. The determinacy of context-free games. J. Symbolic Logic 78 (2013), no. 4, 1115--1134. doi:10.2178/jsl.7804050.

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