## The determinacy of context-free games.(English)Zbl 1349.03038

Summary: 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 $$W(L(\mathcal {A}), L(\mathcal {B}))$$.

### MSC:

 03D05 Automata and formal grammars in connection with logical questions 03E15 Descriptive set theory 03E60 Determinacy principles 91A44 Games involving topology, set theory, or logic
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