Fundamental Properties of Probabilistic Branching-Time Logics

Bc. Jan Křetínský, Ph.D.

Master's thesis

Abstract:

Tato diplomová práce zkoumá pravděpodobnostní logiky větvícího se času, zejména qPCTL a qPCTL*. Soustředíme se na problém splnitelnosti a na existenci axiomatických systémů pro dané logiky. Práce je zčásti založena na spoluautorském článku The satisfiability problem for probabilistic CTL přijatého na konferenci LICS 2008.

Abstract:

This thesis investigates probabilistic branching-time logics, particularly qPCTL and qPCTL*, that arise from CTL and CTL*, respectively, by replacing the $\textsc{EX}$, $\textsc{AX}$, $\textsc{EU}$ and $\textsc{AU}$ operators with their qualitative counterparts $\X^{>0}$, $\X^{=1}$, $\U^{>0}$ and $\U^{=1}$, respectively. We focus on their fundamental properties, such as decidability and complexity of the satisfiability and validity problems, and existence of a sound and complete axiomatic systems for these logics. Firstly, the known results are surveyed, together with the techniques used. Secondly, new results based on a co-authorial article \emph{The satisfiability problem for probabilistic CTL} \cite{LICS,LICSreport} are presented. These results include \EXPTIME-completeness of satisfiability for qPCTL interpreted over Markov chains with countable state space, together with an algorithm yielding finite representations of models for satisfiable formulae. Moreover, the \EXPTIME-hardness proof covers the cases of finite and bounded interpretations, too. In addition, the difference between general and bounded satisfiability is investigated. Furthermore, results on the unrestricted PCTL are presented. The problem whether a given PCTL formula $\varphi$ has a model with branching degree at most $k$, where $k\geq 2$ is an arbitrary but fixed constant, is shown to be highly undecidable. We also show that every satisfiable formula $\varphi$ has a model with branching degree at most $|\varphi| + 2$.

Keywords

branching-time logics, probabilistic CTL (PCTL), qualitative PCTL (qPCTL), PCTL*, qPCTL*, Markov chains, satisfiability, validity, axiomatization

Language used: English

Date on which the thesis was submitted / produced: 25. 5. 2009

Thesis defence

• Date of defence: 2. 7. 2009
• Supervisor: doc. RNDr. Tomáš Brázdil, Ph.D.
• Reader: doc. Mgr. Jan Obdržálek, PhD.

Masaryk University

Faculty of Informatics

Master programme / field:
Informatics / Parallel and Distributed Systems