Qualitative Determinacy and Decidability of Stochastic Games with Signals

We consider the standard model of finite two-person zero-sum stochastic games with signals. We are interested in the existence of almost-surely winning or positively winning strategies, under reachability, safety, Buchi or co-Buchi winning objectives. We prove two qualitative determinacy results. First, in a reachability game either player $1$ can achieve almost-surely the reachability objective, or player 2 can ensure surely the complementary safety objective, or both players have positively winning strategies. Second, in a Buchi game if player 1 cannot achieve almost-surely the Buchi objective, then player 2 can ensure positively the complementary co-Buchi objective. We prove that players only need strategies with finite-memory, whose sizes range from no memory at all to doubly-exponential number of states, with matching lower bounds. Together with the qualitative determinacy results, we also provide fix-point algorithms for deciding which player has an almost-surely winning or a positively winning strategy and for computing the finite memory strategy. Complexity ranges from EXPTIME to 2-EXPTIME with matching lower bounds, and better complexity can be achieved for some special cases where one of the players is better informed than her opponent.