## Paper: The Satisfiability Problem for Probabilistic CTL (at LICS 2008)

**Tomáš Brázdil Vojtech Forejt Jan Kretínský Antonín Kučera**

### Abstract

We study the satisfiability problem for qualitative PCTL(Probabilistic Computation Tree Logic), which is obtained from "ordinary" CTL by replacing the EX, AX, EU, and AU operators with their qualitative counterparts. As opposed to CTL, qualitative PCTL does not have a small model property, and there are even qualitative PCTL formulae which have only infinite-state models. Nevertheless, we show that the satisfiability problem for qualitative PCTL is EXPTIME-complete and we give an exponential-time algorithm which for a given formula computes a finite description of a model (if it exists), or answers "not satisfiable" (otherwise). We also consider the finite satisfiability problem and provide analogous results. That is, we show that the finite satisfiability problem for qualitative PCTLis EXPTIME-complete, and every finite satisfiable formula has a model of an exponential size which can effectively be constructed in exponential time. Finally, we give some results about the quantitative PCTL, where the numerical bounds in probability constraints can be arbitrary rationals between 0 and 1. We prove that the problem whether a given quantitative PCTL formula has a model of the branching degree at most k, wherek > 1 is an arbitrary but fixed constant, is highly undecidable.

### BibTeX

@InProceedings{BrzdilForejtKretins-TheSatisfiabilityPr, author = {Tomáš Brázdil and Vojtech Forejt and Jan Kretínský and Antonín Kučera}, title = { The Satisfiability Problem for Probabilistic CTL}, booktitle = {Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science (LICS 2008)}, year = {2008}, month = {June}, pages = {391--402}, location = {Pittsburgh, PA, USA}, publisher = {IEEE Computer Society Press} }