Abstract. The probabilistic modal μ-calculus pLμ (often called the quantitative μ-calculus) is a generalization of the standard modal μ-calculus designed for expressing properties of probabilistic labeled transition systems. The syntax of pLμ formulas coincides with that of the standard modal μ-calculus. Two equivalent semantics have been studied for pLμ, both assigning to each process-state p a value in [0, 1] representing the probability that the property expressed by the formula will hold in p: a denotational semantics and a game semantics given by means of two player stochastic games. In this paper we extend the logic pLμ with a second conjunction called product, whose semantics interprets the two conjuncts as probabilistically independent events. This extension allows one to encode useful operators, such as the modalities with probability one and with non-zero probability. We provide two semantics for this extended logic: one denotational and one based on a new class of games which we call tree games. The main result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at the first uncountable cardinal.

Abstract. Several notions of bisimulation relations for probabilistic nondeterministic transition systems have been considered in the literature. We study a novel testing-based behavioral equivalence called upper expectation bisimilarity and, using standard results from functional analysis, we develop its coalgebraic and algebraic theory and provide a logical characterization in terms of an expressive probabilistic modal μ-calculus.

Abstract. The probabilistic (or quantitative) modal µ-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS). Two semantics have been studied for this logic, both assigning to every process 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 denotational and the other is a game semantics, specified in terms of two-player stochastic games. The two semantics have been proved to coincide on all finite PLTS's, but the equivalence of the two semantics on arbitrary models has been open in literature. In this paper we prove that the equivalence indeed holds for arbitrary infinite models, and thus our result strengthens the fruitful connection between denotational and game semantics. Our proof adapts the unraveling or unfolding method, a general proof technique for proving result of parity games by induction on their complexity.

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