LIPIcs

We study two-player zero-sum games over infinite-state graphs equipped with ωB and finitary conditions. Our first contribution is about the strategy complexity, i.e the memory required for winning strategies: we prove that over general infinite-state graphs, memoryless strategies are sufficient for finitary Büchi, and finite-memory suffices for finitary parity games. We then study pushdown games with boundedness conditions, with two contributions. First we prove a collapse result for pushdown games with ωB-conditions, implying the decidability of solving these games. Second we consider pushdown games with finitary parity along with stack boundedness conditions, and show that solving these games is EXPTIME-complete

Parity and Streett Games with Costs

We consider two-player games played on finite graphs equipped with costs on edges and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Both conditions generalize the corresponding classical omega-regular conditions and the corresponding finitary conditions. For parity games with costs we show that the first player has positional winning strategies and that determining the winner lies in NP and coNP. For Streett games with costs we show that the first player has finite-state winning strategies and that determining the winner is EXPTIME-complete. The second player might need infinite memory in both games. Both types of games with costs can be solved by solving linearly many instances of their classical variants.Comment: A preliminary version of this work appeared in FSTTCS 2012 under the name "Cost-parity and Cost-Streett Games". The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreements 259454 (GALE) and 239850 (SOSNA

Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

The paper is devoted to several infinite-state Attacker–Defender games with reachability objectives. We prove the undecidability of checking for the existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games. To prove these results, we consider a model of weighted automata operating on infinite words and prove that the universality problem is undecidable for this new class of weighted automata. We show that the universality problem is undecidable by using a non-standard encoding of the infinite Post correspondence problem

Optimality and resilience in parity games

Modeling reactive systems as infinite games has yielded a multitude of results in the fields of program verification and program synthesis. The canonical parity condition, however, neither suffices to express non-functional requirements on the modeled system, nor to capture malfunctions of the deployed system. We address these issues by investigating quantitative games in which the above characteristics can be expressed. Parity games with costs are a variant of parity games in which traversing an edge incurs some nonnegative cost. The cost of a play is the limit superior of the cost incurred between answering odd colors by larger even ones. We extend that model by using integer costs, obtaining parity games with weights, and show that the problem of solving such games is in the intersection of NP and coNP and that it is PTIME-equivalent to the problem of solving energy parity games. We moreover show that Player 0 requires exponential memory to implement a winning strategy in parity games with weights. Further, we show that the problem of determining whether Player 0 can keep the cost of a play below a given bound is EXPTIME-complete for parity games with weights and PSPACE-complete for the special cases of parity games with costs and finitary parity games, i.e., it is harder than solving the game. Thus, optimality comes at a price even in finitary parity games. We further determine the complexity of computing strategies in parity games that are resilient against malfunctions. We show that such strategies can be effectively computed and that this is as hard as solving the game without disturbances. Finally, we combine all these aspects and show that Player 0 can trade memory, cost, and resilience for one another. Furthermore, we show how to compute the possible tradeoffs for a given game.Die Modellierung von reaktiven Systemen durch unendliche Spiele ermöglichte zahlreiche Fortschritte in der Programmverifikation und der Programmsynthese. Die häufig genutzte Paritätsbedingung kann jedoch weder nichtfunktionale Anforderungen ausdrücken, noch Fehlfunktionen des Systems modellieren. Wir betrachten quantitative Spiele in denen diese Merkmale ausgedrückt werden können. Paritätsspiele mit Kosten (PSK) sind eine Variante der Paritätsspiele in denen die Benutzung einer Kante nichtnegative Kosten verursacht. Die Kosten einer Partie sind der Limes Superior der Kosten zwischen ungeraden und den jeweils nächsten größeren geraden Farben. Wir erweitern dieses Modell durch ganzzahlige Kosten zu Paritätsspielen mit Gewichten (PSG). Wir zeigen, dass das Lösen dieser Spiele im Schnitt von NP und coNP liegt, dass es PTIME-äquivalent dazu ist, Energieparitätsspiele zu lösen und dass Spieler 0 exponentiellen Speicher benötigt, um zu gewinnen. Ferner zeigen wir, dass das Problem, zu entscheiden, ob Spieler 0 die Kosten eines Spiels unter einer gegebenen Schranke halten kann, EXPTIME-vollständig für PSG ist, sowie dass es PSPACE-vollständig für die Spezialfälle PSK und finitäre Paritätsspiele (FPS) ist. Optimalität ist also selbst in FPS nicht kostenlos. Außerdem bestimmen wir die Komplexität davon, Strategien in Paritätsspielen zu berechnen, die robust gegenüber Fehlfunktionen sind, zeigen, dass solche Strategien effektiv berechnet werden können und beweisen, dass dies nur linearen Mehraufwand bedeutet. Darüberhinaus kombinieren wir die oben genannten Aspekte, zeigen, dass Spieler 0 Speicher, Kosten und Robustheit gegeneinander eintauschen kann und berechnen die möglichen Kompromisse

Infinite-state games with finitary conditions ∗

We study two-player zero-sum games over infinite-state graphs equipped with ωB and finitary conditions. Our first contribution is about the strategy complexity, i.e the memory required for winning strategies: we prove that over general infinite-state graphs, memoryless strategies are sufficient for finitary Büchi, and finite-memory suffices for finitary parity games. We then study pushdown games with boundedness conditions, with two contributions. First we prove a collapse result for pushdown games with ωB-conditions, implying the decidability of solving these games. Second we consider pushdown games with finitary parity along with stack boundedness conditions, and show that solving these games is EXPTIME-complete