Matteo Mio

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 (PLTSs). 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 denotational and the other is a game semantics, 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 PLTSs 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 modal {\mu}-calculus with independent product, 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 Martins Axiom at an uncountable cardinal.

Measure Properties of Game Tree Languages

We introduce a general method for proving measurability of topologically complex sets by establishing a correspondence between the notion of game tree languages from automata theory and the σ-algebra of \(\mathcal{R}\)-sets, introduced by A. Kolmogorov as a foundation for measure theory. We apply the method to answer positively to an open problem regarding the game interpretation of the probabilistic μ-calculus.

Upper-Expectation Bisimilarity and Łukasiewicz μ-Calculus

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.

Measure properties of regular sets of trees

Abstract We investigate measure theoretic properties of regular sets of infinite trees. As a first result, we prove that every regular set is universally measurable and that every Borel measure on the Polish space of trees is continuous with respect to a natural transfinite stratification of regular sets into ω 1 ranks. We also expose a connection between regular sets and the σ-algebra of R -sets, introduced by A. Kolmogorov in 1928 as a foundation for measure theory. We show that the game tree languages W i , k are Wadge-complete for the finite levels of the hierarchy of R -sets. We apply these results to answer positively an open problem regarding the game interpretation of the probabilistic μ-calculus.

On the Regular Emptiness Problem of Subzero Automata.

Subzero automata is a class of tree automata whose acceptance condition can express probabilistic constraints. Our main result is that the problem of determining if a subzero automaton accepts some regular tree is decidable.

On the Problem of Computing the Probability of Regular Sets of Trees

We consider the problem of computing the probability of regular languages of infinite trees with respect to the natural coin-flipping measure. We propose an algorithm which computes the probability of languages recognizable by \emph{game automata}. In particular this algorithm is applicable to all deterministic automata. We then use the algorithm to prove through examples three properties of measure: (1) there exist regular sets having irrational probability, (2) there exist comeager regular sets having probability

Riesz Modal logic for Markov processes

0

Measure Quantifier in Monadic Second Order Logic

and (3) the probability of \emph{game languages}

Riesz Modal Logic with Threshold Operators

W_{i,k}

Proceedings Tenth International Workshop on Fixed Points in Computer Science

, from automata theory, is