The theory of graph games with omega-regular winning conditions is the foundation for modeling and synthesizing reactive processes. In the case of stochastic reactive processes, the corresponding stochastic graph games have three players, two of them (System and Environment) behaving adversarially, and the third (Uncertainty) behaving probabilistically. We consider two problems for stochastic graph games: the qualitative problem asks for the set of states from which a player can win with probability 1 (almost-sure winning); and the quantitative problem asks for the maximal probability of winning (optimal winning) from each state. We consider omega-regular winning conditions formalized as Muller winning conditions. We present optimal memory bounds for pure (deterministic) almost-sure winning and optimal winning strategies in stochastic graph games with Muller winning conditions. We also present improved memory bounds for randomized almost-sure winning and optimal strategies. We study the complexity of stochastic Muller games and show that the quantitative analysis problem is PSPACE-complete. Our results are relevant in synthesis of stochastic reactive processes.




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