Qualitative Concurrent Games with Imperfect Information

We define a model of games that combines concurrency, imperfect information and stochastic aspects. Those are finite states games in which, at each round, the two players choose, \emph{simultaneously} and \emph{independently}, an action. Then a successor state is chosen accordingly to some fixed probability distribution depending on the previous state and on the pair of actions chosen by the players. Imperfect information is modelled as follows: both players have an equivalence relation over states and, instead of observing the exact state, they only know to which equivalence class it belongs. Therefore, if two partial plays are indistinguishable by some player, he should behave the same in both of them. We consider reachability (does the play eventually visit a final state?) and Buchi objective (does the play visit infinitely often a final state?). A play is won by the first player whenever it satisfies the objective. Our main contribution is to prove that the following problem is $2$-\textsc{ExpTime}-complete: decide whether the first player has a strategy that ensures her to almost-surely win against \emph{any} possible strategy of her adversary. We also precisely characterise those strategies needed by the first player to almost-surely win.