Stochastic Games with Finitary Objectives

The synthesis of a reactive system with respect to an ωregular specification requires the solution of a graph game. Such games have been extended in two natural ways. First, a game graph can be equipped with probabilistic choices between alternative transitions, thus allowing the modeling of uncertain behavior. These are called stochastic games. Second, a liveness specification can be strengthened to require satisfaction within an unknown but bounded amount of time. These are called finitary objectives. We study, for the first time, the combination of stochastic games and finitary objectives. We characterize the requirements on optimal strategies and provide algorithms for computing the maximal achievable probability of winning stochastic games with finitary parity or Streett objectives. Most notably, the set of states from which a player can win with probability 1 for a finitary parity objective can be computed in polynomial time, even though no polynomial-time algorithm is known in the nonfinitary case.