Automata-Based CSL Model Checking
For continuous-time Markov chains, the model-checking problem with respect to continuous-time stochastic logic (CSL) has been introduced and shown to be decidable by Aziz, Sanwal, Singhal and Brayton in 1996. The presented decision procedure, however, has exponential complexity. In this paper, we propose an effective approximation algorithm for full CSL.
The key to our method is the notion of stratified CTMCs with respect to the CSL property to be checked. We present a measure-preservation theorem allowing us to reduce the problem to a transient analysis on stratified CTMCs. The corresponding probability can then be approximated in polynomial time (using uniformization). This makes the present work the centerpiece of a broadly applicable full CSL model checker.
Recently, the decision algorithm by Aziz et al. was shown to be incorrect in general. In fact, it works only for stratified CTMCs. As an additional contribution, our measure-preservation theorem can be used to ensure the decidability for general CTMCs.
KeywordsMarkov Chain Model Check Atomic Proposition State Formula Model Check Problem
Unable to display preview. Download preview PDF.
- 6.Chen, T., Han, T., Katoen, J.P., Mereacre, A.: Quantitative model checking of continuous-time Markov chains against timed automata specifications. In: Twenty-fourth Annual IEEE Symposium on Logic in Computer Science, pp. 309–318. IEEE Comp. Soc., Los Alamitos (2009)Google Scholar
- 8.Grassmann, W.K.: Finding transient solutions in Markovian event systems through randomization. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability, Pure and Applied, vol. 8, pp. 357–371. Marcel Dekker, New York (1991)Google Scholar
- 10.Jansen, D.N.: Erratum to: Model-checking continuous-time Markov chains by Aziz et al. (February 2011), http://arxiv.org/abs/1102.2079v1
- 12.Safra, S.: On the complexity of ω-automata. In: 29th Ann. Symp. on Foundations of Computer Science, pp. 319–327. IEEE Comp. Soc., Los Alamitos (1988)Google Scholar
- 13.Spieler, D.: Model checking of oscillatory and noisy periodic behavior in Markovian population models. Master’s thesis, Saarland University, Saarbrücken (2009), http://alma.cs.uni-sb.de/data/david/mt.pdf
- 15.Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: Symposium on Logic in Computer Science, pp. 332–345. IEEE Comp. Soc., Los Alamitos (1986)Google Scholar
- 16.Zhang, L., Jansen, D.N., Nielson, F., Hermanns, H.: Automata-based CSL model checking (April 2011), http://arxiv.org/abs/1104.4983