Model Checking of Continuous-Time Markov Chains Against Timed Automata Specifications

Abstract

Abstract. We study the verification of a finite continuous-time Markov chain (CTMC) C against a linear real-time specification given as a deterministic timed automaton (DTA) A with finite or Muller acceptance conditions. The central question that we address is: what is the probability of the set of paths of C that are accepted by A, i.e., the likelihood that C satisfies A? It is shown that under finite acceptance criteria this equals the reachability probability in a finite piecewise deterministic Markov process (PDP), whereas for Muller acceptance criteria it coincides with the reachability probability of terminal strongly connected components in such a PDP. Qualitative verification is shown to amount to a graph analysis of the PDP. Reachability probabilities in our PDPs are then characterized as the least solution of a system of Volterra integral equations of the second type and are shown to be approximated by the solution of a system of partial differential equations. For single-clock DTA, this integral equation system can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations. As the coefficients are in fact transient probabilities in CTMCs, this result implies that standard algorithms for CTMC analysis suffice to verify single-clock DTA specifications.1998 ACM Subject Classification: D.2.4.

Almost-Sure Model-Checking of Reactive Timed Automata

“…We already mentioned several related works, but we would like to discuss a bit more the works [17], [19], which we think are the closest to the present paper. In both papers the model is that of CTMCs.…”

“…These properties can be given either as formulas of e.g. the logic CSL or extensions thereof [15], [16], [17], [18], or (deterministic) timed automata [19]. This has led to the development of exact and approximated model-checking algorithms.…”

“…In both papers the model is that of CTMCs. Timing constraints are expressed in the properties, either given as deterministic timed automata [19] or as an extension of CSL called CSL TA [17], which extends CSL with properties given as single-clock deterministic timed automata.…”

Model checking conditional CSL for continuous-time Markov chains

“…There is a rich literature on model checking techniques for CTMCs, see [1,2,[5][6][7][8]. In [5,6], deterministic timed automata (DTA) are used for specifying path properties.…”

“…In [5,6], deterministic timed automata (DTA) are used for specifying path properties. As discussed in [9], nested until CSL path formulas can be expressed in DTA as well, however with a much larger number of states.…”