From Quantity to Quality

In temporal-logic model checking, we verify the correctness of a program with respect to a desired behavior by checking whether a structure that models the program satis es a temporal-logic formula that speci es the behavior. The model-checking problem for the branching-time temporal logic CTL can be solved in linear running time, and model-checking tools for CTL are used successfully in industrial applications. The development of programs that must meet rigid real-time constraints has brought with it a need for real-time temporal logics that enable quantitative reference to time. Early research on real-time temporal logics uses the discrete domain of the integers to model time. Present research on real-time temporal logics focuses on continuous time and uses the dense domain of the reals to model time. There, model checking becomes signi cantly more complicated. For example, the model-checking problem for TCTL, a continuous-time extension of the logic CTL, is PSPACE-complete. In this paper we suggest a reduction from TCTL model checking to CTL model checking. The contribution of such a reduction is twofold. Theoretically, while it has long been known that model-checking methods for untimed temporal logics can be extended quite easily to handle discrete time, it was not clear whether and how untimed methods can handle the reset quanti er of TCTL, which resets a realvalued clock. Practically, our reduction enables anyone who has a tool for CTLmodel checking to use it for TCTL model checking. The TCTL model-checking algorithm that follows from our reduction is in PSPACE, matching the known bound for this problem. In addition, it enjoys the wide distribution of CTL model-checking tools and the extensive and fruitful research e orts and heuristics that have been put into these tools.