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.