Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. The problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas. Depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATLa.ATL and ATLa are interpreted over concurrent game structures. Every state transition of a concurrent game structure results from a choice of moves, one for each player. The players represent individual components and the environment of an open system. Concurrent game structures can capture various forms of synchronous composition for open systems, and if augmented with fairness constraints, also asynchronous composition. Over structures without fairness constraints, the model-checking complexity of ATL is linear in the size of the game structure and length of the formula, and the symbolic model-checking algorithm for CTL extends with few modifications to ATL. Over structures with weak-fairness constraints, ATL model checking requires the solution of 1-pair Rabin games, and can be done in polynomial time. Over structures with strong-fairness constraints, ATL model checking requires the solution of games with Boolean combinations of Buchi conditions, and can be done in PSPACE. In the case of ATLa, the model-checking problem is closely related to the synthesis problem for linear-time formulas, and requires doubly exponential time.

/pdf/alternating-time-temporal-logic-3du0km9tzh.pdf

Alternating-time temporal logic

https://www.cs.le.ac.uk/people/np183/publications/2006/PPS06.pdf

Synthesis of Reactive(1) designs

Infinite games on finitely coloured graphs with applications to automata on infinite trees

We consider turn-based stochastic games on infinite graphs induced by game probabilistic lossy channel systems (GPLCS), the game version of probabilistic lossy channel systems (PLCS). We study games with Buchi (repeated reachability) objectives and almost-sure winning conditions. These games are pure memoryless determined and, under the assumption that the target set is regular, a symbolic representation of the set of winning states for each player can be effectively constructed. Thus, turn-based stochastic games on GPLCS are decidable. This generalizes the decidability result for PLCS-induced Markov decision processes in [10].

/pdf/stochastic-games-with-lossy-channels-4lfbmxienp.pdf

Stochastic games with lossy channels

This paper presents algorithms for the automatic synthesis of real-time controllers by finding a winning strategy for certain games defined by the timed-automata of Alur and Dill. In such games, the outcome depends on the players' actions as well as on their timing. We believe that these results will pave the way for the application of program synthesis techniques to the construction of real-time embedded systems from their specifications.

On the synthesis of discrete controllers for timed systems

Deciding the winner in parity games is in UP ∩ co-UP

In this paper we develop a new algorithm for deciding the winner in parity games, and hence also for the modal µ-calculus model checking. The design and analysis of the algorithm is based on a notion of game progress measures: they are witnesses for winning strategies in parity games. We characterize game progress measures as pre-fixed points of certain monotone operators on a complete lattice. As a result we get the existence of the least game progress measures and a straightforward way to compute them. The worst-case running time of our algorithm matches the best worst-case running time bounds known so far for the problem, achieved by the algorithms due to Browne et al., and Seidl. Our algorithm has better space complexity: it works in small polynomial space; the other two algorithms have exponential worst-case space complexity.

/pdf/small-progress-measures-for-solving-parity-games-4xap7x1b9d.pdf

Small Progress Measures for Solving Parity Games

In this paper we develop a new algorithm for deciding the winner in parity games, and hence also for the modal μ-calculus model checking. The design and analysis of the algorithm is based on a notion of game progress measures: they are witnesses for winning strategies in parity games. We characterize game progress measures as pre-fixed points of certain monotone operators on a complete lattice. As a result we get the existence of the least game progress measures and a straightforward way to compute them. The worst-case running time of our algorithm matches the best worst-case running time bounds known so far for the problem, achieved by the algorithms due to Browne et al., and Seidl. Our algorithm has better space complexity: it works in small polynomial space; the other two algorithms have exponential worst-case space complexity.

Small progress measures for solving parity games

A discrete strategy improvement algorithm is given for constructing winning strategies in parity games, thereby providing also a new solution of the model-checking problem for the modal μ-calculus Known strategy improvement algorithms, as proposed for stochastic games by Hoffman and Karp in 1966, and for discounted payoff games and parity games by Puri in 1995, work with real numbers and require solving linear programming instances involving high precision arithmetic In the present algorithm for parity games these difficulties are avoided by the use of discrete vertex valuations in which information about the relevance of vertices and certain distances is coded An efficient implementation is given for a strategy improvement step Another advantage of the present approach is that it provides a better conceptual understanding and easier analysis of strategy improvement algorithms for parity games However, so far it is not known whether the present algorithm works in polynomial time The long standing problem whether parity games can be solved in polynomial time remains open

A Discrete Strategy Improvement Algorithm for Solving Parity Games

We study perfect-information stochastic parity games. These are two-player nonterminating games which are played on a graph with turn-based probabilistic transitions. A play results in an infinite path and the conflicting goals of the two players are ω-regular path properties, formalized as parity winning conditions. The qualitative solution of such a game amounts to computing the set of vertices from which a player has a strategy to win with probability 1 (or with positive probability). The quantitative solution amounts to computing the value of the game in every vertex, i.e., the highest probability with which a player can guarantee satisfaction of his own objective in a play that starts from the vertex.For the important special case of one-player stochastic parity games (parity Markov decision processes) we give polynomial-time algorithms both for the qualitative and the quantitative solution. The running time of the qualitative solution is O(d · m3/2) for graphs with m edges and d priorities. The quantitative solution is based on a linear-programming formulation.For the two-player case, we establish the existence of optimal pure memoryless strategies. This has several important ramifications. First, it implies that the values of the games are rational. This is in contrast to the concurrent stochastic parity games of de Alfaro et al.; there, values are in general algebraic numbers, optimal strategies do not exist, and e-optimal strategies have to be mixed and with infinite memory. Second, the existence of optimal pure memoryless strategies together with the polynomial-time solution forone-player case implies that the quantitative two-player stochastic parity game problem is in NP ∩ co-NP. This generalizes a result of Condon for stochastic games with reachability objectives. It also constitutes an exponential improvement over the best previous algorithm, which is based on a doubly exponential procedure of de Alfaro and Majumdar for concurrent stochastic parity games and provides only e-approximations of the values.

https://www.dcs.warwick.ac.uk/~mju/Papers/CJH04-SODA.pdf

Marcin Jurdziński

Papers

Deciding the winner in parity games is in UP ∩ co-UP

Small Progress Measures for Solving Parity Games

Small progress measures for solving parity games

A Discrete Strategy Improvement Algorithm for Solving Parity Games

Quantitative stochastic parity games