Abstract. This paper considers algorithms for counterexample generation for (bounded) probabilistic reachability properties in fully probabilistic systems. Finding the strongest evidence (i.e, the most probable path) violating a (bounded) until-formula is shown to be reducible to a single-source (hop-constrained) shortest path problem. Counterexamples of smallest size that are mostly deviating from the required probability bound can be computed by adopting (partially new hopconstrained) k shortest paths algorithms that dynamically determine k.
Abstract-Markov decision processes (MDPs) are often used for modelling distributed systems with probabilistic failure or randomisation. We consider the problem of model repair for MDPs defined as follows: if the MDP fails to satisfy a property, we aim to find new values for the transition probabilities so that the property is guaranteed to hold, while at the same time the cost of repair is minimised. Because solving the MDP repair problem exactly is infeasible, in this paper we focus on approximate solution methods. We first formulate a region-based approach, which yields an interval in which the minimal repair cost is contained. As an alternative, we also consider samplingbased approaches, which are faster but unable to provide lower bounds on the repair cost. We have integrated both methods into the probabilistic model checker PRISM and demonstrated their usefulness in practice using a computer virus case study.
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.
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