Abstract. We consider Markov Decision Processes (MDPs) with mean-payoff parity and energy parity objectives. In system design, the parity objective is used to encode ω-regular specifications, while the mean-payoff and energy objectives can be used to model quantitative resource constraints

We present faster and dynamic algorithms for the following problems arising in probabilistic verification: Computation of the maximal end-component (mec) decomposition of Markov decision processes (MDPs), and of the almost sure winning set for reachability and parity objectives in MDPs. We achieve

to instances of 2-player games and Markov Decision Processes (MDPs) with quantitative winning objectives. QUASY can also be seen as a game solver for quantitative games. Most notable, it can solve lexicographic mean-payoff games with2players, MDPs with mean-payoff objectives, and ergodic MDPs with meanpayoff

, which can be done in polynomial time. For general omega-regular specifications, the solution rests on a new, polynomial-time algorithm for computing optimal strategies in MDPs with mean-payoff parity objectives. We present some experimental results showing optimal systems that were automatically

of the second population. Facing two potentially indistinguishable environments can be easily modeled with a partially observable MDPs. Unfortunately, this model is particularly intractable [3] (e. g. quantitative reachability, safety, and parity objectives, and even qualitative parity objectives * Supported

-state zero-sum stochastic games, with total expected reward objective. They subsume standard finite-state MDPs and Condon’s simple stochastic games and correspond to optimization and game versions of several classic stochastic models, with rewards. Such stochastic models arise naturally as models