The Determinacy of Context-Free Games

Olivier Finkel (1)

1 ELM - Équipe de Logique Mathématique

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 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)).

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

Domaines: Informatique et théorie des jeux [cs.GT]; Logique en informatique [cs.LO]; Logique [math.LO]

Olivier Finkel. The Determinacy of Context-Free Games. 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, 2012, Paris, France. pp.555-566, ⟨10.4230/LIPIcs.STACS.2012.555⟩. ⟨hal-00648462⟩