The Complexity of Partial-observation Stochastic Parity Games With   Finite-memory Strategies
  We consider two-player partial-observation stochastic games on finite-state graphs where player 1 has partial observation and player 2 has perfect observation. The winning condition we study are \omega-regular conditions specified as parity objectives. The qualitative-analysis problem given a partial-observation stochastic game and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability~1 (resp. positive probability). These qualtitative-analysis problems are known to be undecidable. However in many applications the relevant question is the existence of finite-memory strategies, and the qualitative-analysis problems under finite-memory strategies was recently shown to be decidable in 2EXPTIME. We improve the complexity and show that the qualitative-analysis problems for partial-observation stochastic parity games under finite-memory strategies are EXPTIME-complete; and also establish optimal (exponential) memory bounds for finite-memory strategies required for qualitative analysis. 
Krishnendu Chatterjee and Laurent Doyen and Sumit Nain and Moshe Y.   Vardi
cs.LO cs.GT