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

A study of pushdown games

Infinite two-player games are of interest in computer science since they provide an algorithmic framework for the study of nonterminating reactive systems. Usually, an infinite game is specified by an ω-language containing all winning plays for one of the two players or by a game graph and a winning condition on infinite paths through this graph. Many algorithmic results are known for the case of regular specifications and for finite game graphs with winning conditions like parity or Muller conditions capturing regular objectives. The present thesis offers results that also cover a class of nonregular specifications as well as a class of infinite game graphs, namely those specified by pushdown automata, i.e, we consider contextfree specifications and pushdown game graphs with parity or Muller winning conditions. We extend various central results known for regular games to the class of pushdown games. We focus on the following four questions. Firstly, we analyze how the format of a pushdown winning strategy matches the type of the pushdown game specification. Here, we establish a strong connection between the formats of specifications and formats of corresponding winning strategies for several types of contextfree games, but we also exhibit cases of contextfree games where this correspondence fails. Secondly, we investigate delay games with contextfree winning conditions. In such a game one of the players has the possibility to postpone his moves for some time, thus obtaining a lookahead on the moves of the opponent. We clarify whether the winner of a deterministic contextfree delay game can be determined effectively as well as what amount of lookahead is necessary to win such games. Thirdly, we investigate the synthesis problem for distributed systems which consist of several cooperating components communicating with each other and with the environment. A distributed system is specified by an architecture comprising the communication structure of the system. Here, we study both main concepts, that of global and that of local specifications. We offer a complete characterization of the decidable architectures for local specifications, which may be deterministic contextfree or regular. Moreover, we prove that, for global deterministic contextfree specifications, the distributed synthesis problem is undecidable. Finally, we address the problem whether the winner of an infinite game can be already determined after a finite play prefix. Extending results for the case of finite game graphs, we introduce finite-duration parity pushdown games and establish their completeness for solving parity pushdown games. This yields a new reduction method to determine the winner of a pushdown game by solving a reachability game on a finite game graph

A study of pushdown games

Infinite two-player games are of interest in computer science since they provide an algorithmic framework for the study of nonterminating reactive systems. Usually, an infinite game is specified by an ω-language containing all winning plays for one of the two players or by a game graph and a winning condition on infinite paths through this graph. Many algorithmic results are known for the case of regular specifications and for finite game graphs with winning conditions like parity or Muller conditions capturing regular objectives. The present thesis offers results that also cover a class of nonregular specifications as well as a class of infinite game graphs, namely those specified by pushdown automata, i.e, we consider contextfree specifications and pushdown game graphs with parity or Muller winning conditions. We extend various central results known for regular games to the class of pushdown games. We focus on the following four questions. Firstly, we analyze how the format of a pushdown winning strategy matches the type of the pushdown game specification. Here, we establish a strong connection between the formats of specifications and formats of corresponding winning strategies for several types of contextfree games, but we also exhibit cases of contextfree games where this correspondence fails. Secondly, we investigate delay games with contextfree winning conditions. In such a game one of the players has the possibility to postpone his moves for some time, thus obtaining a lookahead on the moves of the opponent. We clarify whether the winner of a deterministic contextfree delay game can be determined effectively as well as what amount of lookahead is necessary to win such games. Thirdly, we investigate the synthesis problem for distributed systems which consist of several cooperating components communicating with each other and with the environment. A distributed system is specified by an architecture comprising the communication structure of the system. Here, we study both main concepts, that of global and that of local specifications. We offer a complete characterization of the decidable architectures for local specifications, which may be deterministic contextfree or regular. Moreover, we prove that, for global deterministic contextfree specifications, the distributed synthesis problem is undecidable. Finally, we address the problem whether the winner of an infinite game can be already determined after a finite play prefix. Extending results for the case of finite game graphs, we introduce finite-duration parity pushdown games and establish their completeness for solving parity pushdown games. This yields a new reduction method to determine the winner of a pushdown game by solving a reachability game on a finite game graph

Playing Pushdown Parity Games in a Hurry

We continue the investigation of finite-duration variants of infinite-duration games by extending known results for games played on finite graphs to those played on infinite ones. In particular, we establish an equivalence between pushdown parity games and a finite-duration variant. This allows us to determine the winner of a pushdown parity game by solving a reachability game on a finite tree

Degrees of Lookahead in Context-free Infinite Games

We continue the investigation of delay games, infinite games in which one player may postpone her moves for some time to obtain a lookahead on her opponent\u27s moves. We show that the problem of determining the winner of such a game is undecidable for deterministic context-free winning conditions. Furthermore, we show that the necessary lookahead to win a deterministic context-free delay game cannot be bounded by any elementary function. Both results hold already for restricted classes of deterministic context-free winning conditions