The theory of graph games with ω-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); the quantitative problem asks for the maximal probability of winning (optimal winning) from each state. We show that for Rabin winning conditions, both problems are in NP. As these problems were known to be NP-hard, it follows that they are NP-complete for Rabin conditions, and dually, coNP-complete for Streett conditions. The proof proceeds by showing that pure memoryless strategies suffice for qualitatively and quantitatively winning stochastic graph games with Rabin conditions. This insight is of interest in its own right, as it implies that controllers for Rabin objectives have simple implementations. We also prove that for every ω-regular condition, optimal winning strategies are no more complex than almost-sure winning strategies.
This research was supported in part by the ONR grant N00014-02-1-0671, the AFOSR MURI grant F49620-00-1-0327, and the NSF grant CCR-0225610.
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ICALP: Automata, Languages and Programming
Chatterjee K, De Alfaro L, Henzinger TA. The complexity of stochastic Rabin and Streett games. In: Vol 3580. Springer; 2005:878-890. doi:10.1007/11523468_71
Chatterjee, K., De Alfaro, L., & Henzinger, T. A. (2005). The complexity of stochastic Rabin and Streett games (Vol. 3580, pp. 878–890). Presented at the ICALP: Automata, Languages and Programming, Springer. https://doi.org/10.1007/11523468_71
Chatterjee, Krishnendu, Luca De Alfaro, and Thomas A Henzinger. “The Complexity of Stochastic Rabin and Streett Games,” 3580:878–90. Springer, 2005. https://doi.org/10.1007/11523468_71.
K. Chatterjee, L. De Alfaro, and T. A. Henzinger, “The complexity of stochastic Rabin and Streett games,” presented at the ICALP: Automata, Languages and Programming, 2005, vol. 3580, pp. 878–890.
Chatterjee K, De Alfaro L, Henzinger TA. 2005. The complexity of stochastic Rabin and Streett games. ICALP: Automata, Languages and Programming, LNCS, vol. 3580, 878–890.
Chatterjee, Krishnendu, et al. The Complexity of Stochastic Rabin and Streett Games. Vol. 3580, Springer, 2005, pp. 878–90, doi:10.1007/11523468_71.