Formats of Winning Strategies for Six Types of Pushdown Games

The solution of parity games over pushdown graphs (Walukiewicz '96) was the first step towards an effective theory of infinite-state games. It was shown that winning strategies for pushdown games can be implemented again as pushdown automata. We continue this study and investigate the connection between game presentations and winning strategies in altogether six cases of game arenas, among them realtime pushdown systems, visibly pushdown systems, and counter systems. In four cases we show by a uniform proof method that we obtain strategies implementable by the same type of pushdown machine as given in the game arena. We prove that for the two remaining cases this correspondence fails. In the conclusion we address the question of an abstract criterion that explains the results

Tree Automata Make Ordinal Theory Easy

We give a new simple proof of the decidability of the First Order Theory of $(w^{w^i},+)$ and the Monadic Second Order Theory of $(w^i,<)$, improving the complexity in both cases. Our algorithm is based on tree automata and a new representation of (sets of) ordinals by (infinite) trees

Uniform solution of parity games on prefix-recognizable graphs

Walukiewicz gave in 1996 a solution for parity games on pushdown graphs: he proved the existence of pushdown strategies and determined the winner with an EX-PTIME procedure. We give a new presentation and a new algorithmic proof of these results, obtain a uniform solution for parity games (independent of their initial configu-ration), and extend the results to prefix-recognizable graphs. The winning regions of the players are proved to be effectively regular, and winning strategies are computed.

Jeux sur des graphes d'automates à pile et leurs extensions

On considère des jeux à deux joueurs sur des familles de graphes infinis.Notre but est de déterminer le gagnant et de calculer une stratégie gagnante. Nous avons considéré différentes conditions de gain : accessibilité, Büchi (récurrence), Sigma3, parité, et différentes classes de graphes depuis les graphes de transition des automates à pile jusqu'aux graphes de la hiérarchie de Caucal et aux automates à pile d'ordre supérieur. Deux types de méthodes ont été proposées : une approche symbolique fondée sur des automates finis, et des techniques de jeu-simulation. L'approche symbolique permet de représenter et de manipuler des ensembles infinis de configurations. La jeu-simulation consiste à réduire un jeu donné à un autre jeu plus simple que l'on sait résoudre, puis d'en déduire le gagnant et une stratégie gagnante dans le jeu initial. À chaque fois des algorithmes ont été donnés pour calculer le gagnant et une stratégie gagnante (à coup sûr).RENNES1-BU Sciences Philo (352382102) / SudocSudocFranceF

The complexity of games on higher order pushdown automata

Abstract. We prove an n-exptime lower bound for the problem of deciding the winner in a reachability game on Higher Order Pushdown Automata (HPDA) of level n. This bound matches the known upper bound for parity games on HPDA. As a consequence the µ-calculus model checking over graphs given by n-HPDA is n-exptime complete.

Solving Pushdown Games with a Sigma_3 Winning Condition

We study infinite two-player games over pushdown graphs with a winning condition that refers explicitly to the infinity of the game graph: A play is won by player 0 if some vertex is visited infinity often during the play. We show that the set of winning plays is a proper Sigma-3-set in the Borel hierarchy, thus transcending the Boolean closure of Sigma-2-sets which arises with the standard automata theoretic winning conditions (such as the Muller, Rabin, or parity condition). We also show that this Sigma-3-game over pushdown graphs can be solved effectively (by a computation of the winning region of player 0 and his memoryless winning strategy). This seems to be a first example of an effectively solvable game beyond the second level of the Borel hierarchy