Infinite games recognized by 2-stack visibly pushdown automata
چکیده
Two-player infinite games have been extensively studied in Descriptive Set Theory in the past several decades, accompanied by a celebrated result due to Martin that all Gale-Stewart games with Borel winning conditions are determined. In this paper, we are interested in the following problem: if the winning set X is given as X = L(M), where L(M) is the language recognized by some kind of machine M, whether the game G(X) is determined or not, and if one of the two players has a winning strategy, whether the strategy is computable or not. Büchi and Landweber (1969) first studied the Gale-Stewart game G(X), where X is an ωregular language accepted by a Büchi automaton. They showed that one can effectively decide the winner of G(X) and the winning strategy can be constructed by a finite transducer. Thomas (1995) asked whether it can be extended to deterministic context-free ω-languages, which are accepted by Muller pushdown automata. Walukiewicz (2001) presented a positive answer by showing that one can effectively construct winning strategies in parity games played on pushdown graphs. This result implies that the Gale-Stewart games over deterministic context-free ω-languages are determined with computable winning strategies. Finkel (2001) studied the cases where the winning sets are ω-languages accepted by nondeterministic pushdown automata and showed that in such games, it is undecidable to determine which player has a winning strategy. Finkel (2013) also proved that the determinacy over the class of context-free ω-languages is equivalent to the determinacy over the effective class of analytic sets. Research also extends to infinite games related with some variants of pushdown automata. Well-known variants of pushdown automata include, extensions on the stacks, such as higher-order stacks, multi-stacks, and restrictions on the input alphabet, such as visibly input. As far as we know, two-player infinite games recognized by 2-stack visibly pushdown automata have not been studied up to now. In this paper, we mainly treat 2-visibly pushdown automata with Boolean combination of Σ1 accepting condition. The input alphabet of a 2-stack visibly pushdown automaton is partitioned into push, pop alphabet for each stack separately, and internal alphabet, which decide its visible actions. We show that there exists an infinite game recognized by 2-stack visibly pushdown automata in which Player II has a winning strategy but no computable one.