On the Satisfiability of Some Simple Probabilistic Logics

This paper shows that the satisfiability problems for a bounded fragment of probabilistic CTL (called bounded PCTL) and an extension of the modal μ-calculus with probabilistic quantification over next-modalities (called PμTL) are decidable. For bounded PCTL we provide an NEXP-TIME-algorithm for the satisfiability problem and show that the logic has a small model property where the model size is independent from the probability bounds in the formula. We show that the satisfiability problem of a simple sub-logic of bounded PCTL is PSPACE-complete. We prove that PμTL has a small model property and that a decision procedure using 2 player parity games can be employed for the satisfiability problem of PμTL. These results imply that PμTL and qualitative PCTL formulas with only thresholds >0 and =1—are incomparable. We also establish that—in contrast to PCTL—every satisfiable PμTL-formula has a rational model, a model with rational probabilities only.