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@inproceedings{771768,
author = {Brázdil, Tomáš and Forejt, Vojtěch and Křetínský, Jan and Kučera, Antonín},
address = {Los Alamitos, California},
booktitle = {23rd IEEE Symposium on Logic in Computer Science (LICS 2008), 24-27 June 2008, Pittsburgh, USA, Proceedings},
keywords = {Stochastic systems; branching-time temporal logics},
language = {eng},
location = {Los Alamitos, California},
isbn = {978-0-7695-3183-0},
pages = {391-402},
publisher = {IEEE Computer Society},
title = {The Satisfiability Problem for Probabilistic CTL},
year = {2008}
}
TY - JOUR
ID - 771768
AU - Brázdil, Tomáš - Forejt, Vojtěch - Křetínský, Jan - Kučera, Antonín
PY - 2008
TI - The Satisfiability Problem for Probabilistic CTL
PB - IEEE Computer Society
CY - Los Alamitos, California
SN - 9780769531830
KW - Stochastic systems
KW - branching-time temporal logics
N2 - 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 X>0, X=1, U>0, and U=1, respectively. 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 PCTL is 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, where k > 1 is an arbitrary but fixed constant, is highly undecidable. We also show that every satisfiable formula F has a model with branching degree at most |F| + 2. However, this does not yet imply the undecidability of the satisfiability problem for quantitative PCTL, and we in fact conjecture the opposite.
ER -
BRÁZDIL, Tomáš, Vojtěch FOREJT, Jan KŘETÍNSKÝ and Antonín KUČERA. The Satisfiability Problem for Probabilistic CTL. In \textit{23rd IEEE Symposium on Logic in Computer Science (LICS 2008), 24-27 June 2008, Pittsburgh, USA, Proceedings}. Los Alamitos, California: IEEE Computer Society, 2008, p.~391-402, 10 pp. ISBN~978-0-7695-3183-0.
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