PROBABILISTIC MODAL mu-CALCULUS WITH INDEPENDENT PRODUCT
The probabilistic modal mu-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS's). Two equivalent semantics have been studied for this logic, both assigning to each state a value in the interval [0, 1] representing the probability that the property expressed by the formula holds at the state. One semantics is de no tat ion a l and the other is a g a m e se m antics, specified in terms of two- player stochastic parity games. A shortcoming of the probabilistic modal mu-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL). To address this limitation we extend the logic with a new pair of operators: independent product and coproduct. The resulting logic, called probabilistic mod a l mu-calculus within de pendent pro duct, can encode many properties of interest and subsumes the qualitative fragment of PCTL. The main contribution of this paper is the definition of an appropriate game semantics for this extended probabilistic mu-calculus. This relies on the definition of a new class of games which generalize standard two- player stochastic (parity) games by allowing a play to be split into concurrent subplays, each continuing their evolution independently. Our main technical result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at an uncountable cardinal.
modal mu-calculus game semantics probabilistic temporal logic two-player stochastic games