Extended Directed Search for Probabilistic Timed Reachability
Current numerical model checkers for stochastic systems can efficiently analyse stochastic models. However, the fact that they are unable to provide debugging information constrains their practical use. In precursory work we proposed a method to select diagnostic traces, in the parlance of functional model checking commonly referred to as failure traces or counterexamples, for probabilistic timed reachability properties on discrete-time and continuous-time Markov chains. We applied directed explicit-state search algorithms, like Z ∗ , to determine a diagnostic trace which carries large amount of probability. In this paper we extend this approach to determining sets of traces that carry large probability mass, since properties of stochastic systems are typically not violated by single traces, but by collections of those. To this end we extend existing heuristics guided search algorithms so that they select sets of traces. The result is provided in the form of a Markov chain. Such diagnostic Markov chains are not just essential tools for diagnostics and debugging but, they also allow the solution of timed reachability probability to be approximated from below. In particular cases, they also provide real counterexamples which can be used to show the violation of the given property. Our algorithms have been implemented in the stochastic model checker PRISM. We illustrate the applicability of our approach using a number of case studies.
KeywordsMarkov Chain Model Check Target State Heuristic Function State Transition Graph
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- 1.Holzmann, G.J.: The Spin Model Checker: Primer and Reference Manual. Addison Wesley, Reading (2003)Google Scholar
- 4.Pearl, J.: Heuristics – Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, Reading (1986)Google Scholar
- 10.Katoen, J.P., Khattri, M., Zapreev, I.S.: A markov reward model checker. Qest 0, 243–244 (2005)Google Scholar
- 13.Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P.: Model-checking algorithms for continuous-time Markov chains. IEEE Transions on Software Engineering 29 (2003)Google Scholar
- 17.Goodrich, M., Tamassia, R.: Data Structures and Algorithms in Java, 2nd edn. John Wiley & Sons, Inc., New York (2000)Google Scholar
- 18.JDSL Web Page: http://www.cs.brown.edu/cgc/jdsl/
- 19.Aljazzar, H., Leue, S.: Extended directed search for probabilistic timed reachability. Technical Report soft-06-03, Chair for Software Engineering, University of Konstanz, Gemany (2006), URL: http://www.inf.uni-konstanz.de/soft/research/publications/pdf/soft-06-03.pdf
- 20.Haverkort, B.R., Hermanns, H., Katoen, J.P.: On the use of model checking techniques for dependability evaluation. In: SRDS, pp. 228–237 (2000)Google Scholar