Combinations of Qualitative Winning for Stochastic Parity Games
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We study Markov decision processes and turn-based stochastic games with parity conditions. There are three qualitative winning criteria, namely, sure winning, which requires all paths must satisfy the condition, almost-sure winning, which requires the condition is satisfied with probability 1, and limit-sure winning, which requires the condition is satisfied with probability arbitrarily close to 1. We study the combination of these criteria for parity conditions, e.g., there are two parity conditions one of which must be won surely, and the other almost-surely. The problem has been studied recently by Berthon et. al for MDPs with combination of sure and almost-sure winning, under infinite-memory strategies, and the problem has been established to be in NP ∩ coNP. Even in MDPs there is a difference between finite-memory and infinite-memory strategies. Our main results for combination of sure and almost-sure winning are as follows: (a) we show that for MDPs with finite-memory strategies the problem lie in NP ∩ coNP; (b) we show that for turn-based stochastic games the problem is coNP-complete, both for finite-memory and infinite-memory strategies; and (c) we present algorithmic results for the finite-memory case, both for MDPs and turn-based stochastic games, by reduction to non-stochastic parity games. In addition we show that all the above results also carry over to combination of sure and limit-sure winning, and results for all other combinations can be derived from existing results in the literature. Thus we present a complete picture for the study of combinations of qualitative winning criteria for parity conditions in MDPs and turn-based stochastic games.
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