State Equivalences for Rectangular Hybrid Automata

Three natural equivalence relations on the in nite state space of a hybrid automaton are language equivalence, simulation equivalence, and bisimulation equivalence. When one of these equivalence relations has a nite quotient, certain model checking and controller synthesis problems are decidable. When bounds on the number of equivalence classes are obtained, bounds on the running times of model checking and synthesis algorithms follow as corollaries. We characterize the time-abstract versions of these equivalence relations on the state spaces of rectangular hybrid automata (RHA), in which each continuous variable is a clock with bounded drift. These automata are useful for modeling communications protocols with drifting local clocks, and for the conservative approximation of more complex hybrid systems. Of our two main results, one has positive implications for automatic veri cation, and the other has negative implications. On the positive side, we nd that the ( nite) language equivalence quotient for RHA is coarser than was previously known by a multiplicative exponential factor. On the negative side, we show that simulation equivalence for RHA is equality (which obviously has an in nite quotient). Our main positive result is established by analyzing a subclass of timed automata, called one-sided timed automata (OTA), for which the language equivalence quotient is coarser than for the class all of timed automata. An exact characterization of language equivalence for OTA requires a distinction between synchronous and asynchronous definitions of (bi)simulation: if time actions are silent, then the induced quotient for OTA is coarser than if time actions are visible. A preliminary version of this paper appeared in the Proceedings of the Seventh International Conference on Concurrency Theory (CONCUR 96), Lecture Notes in Computer Science 1119, Springer-Verlag, 1996, pp. 530{545. Supported in part by the O ce of Naval Research Young Investigator award N00014-95-1-0520, by the National Science Foundation CAREER award CCR-9501708, by the National Science Foundation grant CCR-9504469, by the Air Force O ce of Scienti c Research contract F49620-93-1-0056, and by the Advanced Research Projects Agency grant NAG2-892 Supported by the U.S. Army Research O ce through the Mathematical Sciences Institute of Cornell University, Contract Number DAAL03-91-C-0027.