Probabilistic Mu-Calculus: Decidability and Complete Axiomatization

We introduce a version of the probabilistic mu-calculus (PMC) built on top of a probabilistic modal logic that allows encoding n-ary inequational conditions on transition probabilities. PMC extends previously studied calculi and we prove that, despite its expressiveness, it enjoys a series of good meta-properties. Firstly, we prove the decidability of satisfiability checking by establishing the small model property. An algorithm for deciding the satisfiability problem is developed. As a second major result, we provide a complete axiomatization for the alternation-free fragment of PMC. The completeness proof is innovative in many aspects combining various techniques from topology and model theory

Alternation-free weighted mu-calculus : decidability and completeness

In this paper we introduce WMC, a weighted version of the alternation-free modal mu-calculus for weighted transition systems. WMC subsumes previously studied weighted extensions of CTL and resembles previously proposed time-extended versions of the modal mu-calculus. We develop, in addition, a symbolic semantics for WMC and demonstrate that the notion of satisfiability coincides with that of symbolic satisfiability. This central result allows us to prove two major meta-properties of WMC. The first is decidability of satisfiability for WMC. In contrast to the classical modal mu-calculus, WMC does not possess the finite model-property. Nevertheless, the finite model property holds for the symbolic semantics and decidability readily follows; and this contrasts to resembling logics for timed transitions systems for which satisfiability has been shown undecidable. As a second main contribution, we provide a complete axiomatization, which applies to both semantics. The completeness proof is non-standard, since the logic is non-compact, and it involves the notion of symbolic models