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Recurrence and Transience for Probabilistic Automata
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... In a context of ωregular specifications for infinite execution sequences, the classical Büchi condition, or repeated liveness condition, asks that an accepting state is visited infinitely often. In this paper, we show that in a probabilistic context it is relevant to strengthen this infinitely oft ..."
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In a context of ωregular specifications for infinite execution sequences, the classical Büchi condition, or repeated liveness condition, asks that an accepting state is visited infinitely often. In this paper, we show that in a probabilistic context it is relevant to strengthen this infinitely often condition. An execution path is now accepting if the proportion of time spent on an accepting state does not go to zero as the length of the path goes to infinity. We introduce associated notions of recurrence and transience for nonhomogeneous finite Markov chains and study the computational complexity of the associated problems. As Probabilistic Büchi Automata (PBA) have been an attempt to generalize Büchi automata to a probabilistic context, we define a class of Constrained Probabilistic Automata with our new accepting condition on runs. The accepted language is defined by the requirement that the measure of the set of accepting runs is positive (probable semantics) or equals 1 (almostsure semantics). In contrast to the PBA case, we prove that the emptiness problem for the language of a constrained probabilistic Büchi automaton with the probable semantics is decidable.
PerfectInformation Stochastic MeanPayoff Parity Games ⋆,⋆⋆
"... Abstract The theory of graph games is the foundation for modeling and synthesizing reactive processes. In the synthesis of stochastic processes, we use 2 1 2player games where some transitions of the game graph are controlled by two adversarial players, the System and the Environment, and the other ..."
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Abstract The theory of graph games is the foundation for modeling and synthesizing reactive processes. In the synthesis of stochastic processes, we use 2 1 2player games where some transitions of the game graph are controlled by two adversarial players, the System and the Environment, and the other transitions are determined probabilistically. We consider 2 1player games where the objec2 tive of the System is the conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a meanpayoff condition). We establish that the problem of deciding whether the System can ensure that the probability to satisfy the meanpayoff parity objective is at least a given threshold is in NP∩coNP, matching the best known bound in the special case of 2player games (where all transitions are deterministic). We present an algorithm running in timeO(d·n 2d ·MeanGame) to compute the set of almostsure winning states from which the objective can be ensured with probability 1, where n is the number of states of the game, d the number of priorities of the parity objective, and MeanGame is the complexity to compute the set of almostsure winning states in 2 1 2player meanpayoff games. Our results are useful in the synthesis of stochastic reactive systems with both functional requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective). 1