Probabilistic modal {\mu}-calculus with independent product

TL;DR

This paper extends the probabilistic modal μ-calculus with independent product operators to enhance expressiveness, providing a new game semantics based on concurrent subplays, and proves the equivalence of denotational and game semantics.

Contribution

It introduces a new logic with independent product operators and develops a novel game semantics involving concurrent subplays, establishing their semantic equivalence.

Findings

The extended logic can encode properties beyond the original probabilistic μ-calculus.

A new class of games with concurrent subplays is defined for semantics.

The equivalence of denotational and game semantics is proven.

Abstract

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 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 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 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 Martin's Axiom at an uncountable cardinal.