The Determinacy of Context-Free Games

TL;DR

This paper explores the determinacy of certain infinite games defined by automata, showing their equivalence to large cardinal assumptions and constructing automata with models where these games are either determined or not.

Contribution

It establishes the equivalence between automata-based game determinacy and large cardinal hypotheses, and constructs automata illustrating models with different determinacy outcomes.

Findings

Determinacy of Gale-Stewart games with real-time 1-counter B"uchi automata is equivalent to effective analytic game determinacy.

Determinacy of Wadge games between omega-languages accepted by 1-counter B"uchi automata is equivalent to effective analytic Wadge determinacy.

Constructs automata demonstrating models where Wadge games are either determined or not, depending on set-theoretic assumptions.

Abstract

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by real-time 1-counter B\"uchi 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 B\"uchi 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 B\"uchi automaton A and a B\"uchi 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)).