The Observational Power of Clocks

We develop a theory of equivalences for timed systems. Two systems are equivalent i external observers cannot observe di erences in their behavior. The notion of equivalence depends, therefore, on the distinguishing power of the observers. The power of an observer to measure time results in untimed, clock, and timed equivalences: an untimed observer cannot measure the time di erence between events; a clock observer uses a clock to measure time di erences with nite precision; a timed observer is able to measure time di erences with arbitrary precision. We show that the distinguishing power of clock observers grows with the number of observers, and approaches, in the limit, the distinguishing power of a timed observer. More precisely, given any equivalence for untimed systems, two timed systems are k-clock congruent, for a nonnegative integer k, i their compositions with every environment that uses k clocks are untimed equivalent. Both k-clock bisimulation congruence and k-clock trace congruence form strict decidable hierarchies that converge towards the corresponding timed equivalences. Moreover, k-clock bisimulation congruence and k-clock trace congruence provide an adequate and abstract semantics for branching-time and linear-time logics with k clocks. Our results impact on the veri cation of timed systems in two ways. First, our decision procedure for k-clock bisimulation congruence leads to a new, symbolic, decision procedure for timed bisimilarity. Second, timed trace equivalence is known to be undecidable. If the number of environment clocks is bounded, however, then our decision procedure for k-clock trace congruence allows the veri cation of timed systems in a trace model. A preliminary version of this paper appeared in the Proceedings of the International Conference on Concurrency Theory (CONCUR 94), Lecture Notes in Computer Science 836, Springer-Verlag, 1994, pp. 162{177. AT&T Bell Laboratories, Murray Hill, NJ. Department of Computer Science, University of Crete, Greece. Supported in part by the BRA ESPRIT project REACT. Department of Computer Science, Cornell University, Ithaca, NY. Supported in part by the National Science Foundation under grant CCR-9200794 and by the United States Air Force O ce of Scienti c Research under contract F49620-93-1-0056.