Extended Directed Search for Probabilistic Timed Reachability

Abstract

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 work 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.

Counterexamples in Probabilistic Model Checking

“…In our view, the main advantage of our approach is the systematic characterization of generating counterexamples in terms of shortest path problems. Recently, [4] generalizes the heuristic approach to obtain failure subgraphs, i.e., counterexamples. To our knowledge, smallest counterexamples have not been considered yet.…”

“…In probabilistic model checking, however, counterexample generation is almost not developed; notable exception is the recent heuristic search algorithm for CTMCs and DTMCs [3,4] that works under the assumption that the model is unknown. Instead, we consider a setting in which it has already been established that a certain state refutes a given property.…”

Significant Diagnostic Counterexamples in Probabilistic Model Checking

“…Presently, some work has been done on counterexample generation techniques for different variants of probabilistic models (Discrete Markov chains and Continues Markov chains) [AHL05,AL06,HK07a,HK07b]. In our terminology, these works consider witnesses consisting of a single finite path.…”

Counterexamples in Probabilistic LTL Model Checking for Markov Chains

“…We have also pointed out how such a counterexample can be utilised to find an error in the system. Aljazzar and Leue [2] propose solutions for counterexamples in probabilistic model checking with respect to timed probabilistic reachability properties in Markov chains. Han and Katoen [12] and Wimmer et al [23] present algorithms computing counterexamples for model checking PCTL (probabilistic CTL [14]) formulas in Markov chains.…”