Quantitative Solution of Omega-Regular Games

摘要

We consider two-player games played for an infinite number of rounds, with omega-regular winning conditions. The games may be concurrent, in that the players choose their moves simultaneously and independently, and probabilistic, in that the moves determine a probability distribution for the successor state. We introduce quantitative game mu-ca1cu1us, and we show that the maximal probability of winning such games can be expressed as the fixpoint formulas in this calculus. We develop the arguments both for deterministic and for probabilistic concurrent games; as a special case, we solve probabilistic turn- based games with omega-regular winning conditions, which was also open. We also characterize the optimality, and the memory requirements, of the winning strategies. In particular, we show that while memoryless strategies suffice for winning games with safety and reachability conditions, Buechi conditions require the use of strategies with infinite memory. The existence of optimal strategies, as opposed to xi-optimal, is only guaranteed in games with safety winning conditions.