Playing Muller Games in a Hurry

This work studies the following question: can plays in a Muller game be stopped after a finite number of moves and a winner be declared. A criterion to do this is sound if Player 0 wins an infinite-duration Muller game if and only if she wins the finite-duration version. A sound criterion is presented that stops a play after at most 3^n moves, where n is the size of the arena. This improves the bound (n!+1)^n obtained by McNaughton and the bound n!+1 derived from a reduction to parity games

The Church Problem for Countable Ordinals

A fundamental theorem of Buchi and Landweber shows that the Church synthesis problem is computable. Buchi and Landweber reduced the Church Problem to problems about &#969;-games and used the determinacy of such games as one of the main tools to show its computability. We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length. We prove that determinacy and decidability parts of the Bu}chi and Landweber theorem hold for all countable ordinals and that its full extension holds for all ordinals < \omega\^\omega

The Church Synthesis Problem with Parameters

For a two-variable formula &psi;(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of an operator Y=F(X) such that &psi;(X,F(X)) is universally valid over Nat. B\"{u}chi and Landweber proved that the Church synthesis problem is decidable; moreover, they showed that if there is an operator F that solves the Church Synthesis Problem, then it can also be solved by an operator defined by a finite state automaton or equivalently by an MLO formula. We investigate a parameterized version of the Church synthesis problem. In this version &psi; might contain as a parameter a unary predicate P. We show that the Church synthesis problem for P is computable if and only if the monadic theory of is decidable. We prove that the B\"{u}chi-Landweber theorem can be extended only to ultimately periodic parameters. However, the MLO-definability part of the B\"{u}chi-Landweber theorem holds for the parameterized version of the Church synthesis problem

Extracting winning strategies in update games

This paper investigates algorithms for extracting winning strategies in two-player games played on nite graphs. We focus on a special class of games called update games. We present a procedure for extracting winning strategies in update games by constructing strategies explicitly. This is based on an algorithm that solves update games in quadratic time. We also show that solving update games with a bounded number of nondeterministic nodes takes linear time

Energy Parity Games

Energy parity games are infinite two-player turn-based games played on weighted graphs. The objective of the game combines a (qualitative) parity condition with the (quantitative) requirement that the sum of the weights (i.e., the level of energy in the game) must remain positive. Beside their own interest in the design and synthesis of resource-constrained omega-regular specifications, energy parity games provide one of the simplest model of games with combined qualitative and quantitative objective. Our main results are as follows: (a) exponential memory is necessary and sufficient for winning strategies in energy parity games; (b) the problem of deciding the winner in energy parity games can be solved in NP \cap coNP; and (c) we give an algorithm to solve energy parity by reduction to energy games. We also show that the problem of deciding the winner in energy parity games is polynomially equivalent to the problem of deciding the winner in mean-payoff parity games, while optimal strategies may require infinite memory in mean-payoff parity games. As a consequence we obtain a conceptually simple algorithm to solve mean-payoff parity games

Learning-Based Synthesis of Safety Controllers

We propose a machine learning framework to synthesize reactive controllers for systems whose interactions with their adversarial environment are modeled by infinite-duration, two-player games over (potentially) infinite graphs. Our framework targets safety games with infinitely many vertices, but it is also applicable to safety games over finite graphs whose size is too prohibitive for conventional synthesis techniques. The learning takes place in a feedback loop between a teacher component, which can reason symbolically about the safety game, and a learning algorithm, which successively learns an overapproximation of the winning region from various kinds of examples provided by the teacher. We develop a novel decision tree learning algorithm for this setting and show that our algorithm is guaranteed to converge to a reactive safety controller if a suitable overapproximation of the winning region can be expressed as a decision tree. Finally, we empirically compare the performance of a prototype implementation to existing approaches, which are based on constraint solving and automata learning, respectively

Positional Determinacy of Games with Infinitely Many Priorities

We study two-player games of infinite duration that are played on finite or infinite game graphs. A winning strategy for such a game is positional if it only depends on the current position, and not on the history of the play. A game is positionally determined if, from each position, one of the two players has a positional winning strategy. The theory of such games is well studied for winning conditions that are defined in terms of a mapping that assigns to each position a priority from a finite set. Specifically, in Muller games the winner of a play is determined by the set of those priorities that have been seen infinitely often; an important special case are parity games where the least (or greatest) priority occurring infinitely often determines the winner. It is well-known that parity games are positionally determined whereas Muller games are determined via finite-memory strategies. In this paper, we extend this theory to the case of games with infinitely many priorities. Such games arise in several application areas, for instance in pushdown games with winning conditions depending on stack contents. For parity games there are several generalisations to the case of infinitely many priorities. While max-parity games over omega or min-parity games over larger ordinals than omega require strategies with infinite memory, we can prove that min-parity games with priorities in omega are positionally determined. Indeed, it turns out that the min-parity condition over omega is the only infinitary Muller condition that guarantees positional determinacy on all game graphs

Finite-state Strategies in Delay Games (full version)

What is a finite-state strategy in a delay game? We answer this surprisingly non-trivial question by presenting a very general framework that allows to remove delay: finite-state strategies exist for all winning conditions where the resulting delay-free game admits a finite-state strategy. The framework is applicable to games whose winning condition is recognized by an automaton with an acceptance condition that satisfies a certain aggregation property. Our framework also yields upper bounds on the complexity of determining the winner of such delay games and upper bounds on the necessary lookahead to win the game. In particular, we cover all previous results of that kind as special cases of our uniform approach