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Verification of Safety and Liveness Properties of Metric Transition Systems

We consider verification problems for transition systems enriched with a metric structure. We believe that these metric transition systems are particularly suitable for the analysis of cyber-physical systems in which metrics can be naturally defined on the numerical variables of the embedded software and on the continuous states of the physical environment. We consider verification of bounded and unbounded safety properties, as well as bounded liveness properties. The transition systems we consider are nondeterministic, finitely branching, and with a finite set of initial states. Therefore, bounded safety/liveness properties can always be verified by exhaustive exploration of the system trajectories. However, this approach may be intractable in practice, as the number of trajectories usually grows exponentially with respect to the considered bound. Furthermore, since the system we consider can have an infinite set of states, exhaustive exploration cannot be used for unbounded safety verification. For bounded safety properties, we propose an algorithm which combines exploration of the system trajectories and state space reduction using merging based on a bisimulation metric. The main novelty compared to an algorithm presented recently by Lerda et al. [2008] consists in introducing a tuning parameter that improves the performance drastically. We also establish a procedure that allows us to prove unbounded safety from the result of the bounded safety algorithm via a refinement step. We then adapt the algorithm to handle bounded liveness verification. Finally, the effectiveness of the approach is demonstrated by applying it to the analysis of implementations of an embedded control loop.