Three natural equivalence relations on the infinite state space of a hybrid automaton are language equivalence, simulation equivalence, and bisimulation equivalence. When one of these equivalence relations has a finite 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 verification, and the other has negative implications. On the positive side, we find that the (finite) 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 infinite 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 of all 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 (but not their durations) are visible.
- Equivalence Relation
- Model Check
- Fractional Part
- Discrete Action
- Hybrid Automaton
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This research was supported in part by ONR Young Investigator award N00014-95-1-0520, by NSF CAREER award CCR-9501708, by NSF grant CCR-9504469, by Air Force Office of Scientific Research contract F49620-93-1-0056, by ARPA grant NAG2-892, and by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, Contract Number DAAL03-91-C-0027.
This is a preview of subscription content, access via your institution.
Unable to display preview. Download preview PDF.
R. Alur, C. Courcoubetis, N. Halbwachs, T.A. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138:3–34, 1995.
R. Alur, C. Courcoubetis, T.A. Henzinger, and P.-H. Ho. Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In R.L. Grossman, A. Nerode, A.P. Ravn, and H. Rischel, editors, Hybrid Systems I, Lecture Notes in Computer Science 736, pages 209–229. Springer-Verlag, 1993.
R. Alur and D.L. Dill. A theory of timed automata. Theoretical Computer Science, 126:183–235, 1994.
R. Alur, T.A. Henzinger, and P.-H. Ho. Automatic symbolic verification of embedded systems. IEEE Transactions on Software Engineering, 22(3):181–201, 1996.
S. Bensalem, A. Bouajjani, C. Loiseaux, and J. Sifakis. Property-preserving simulations. In G. von Bochmann and D.K. Probst, editors, CAV 92: Computer-aided Verification, Lecture Notes in Computer Science 663, pages 260–273. Springer-Verlag, 1992.
M.C. Browne, E.M. Clarke, and O. Grümberg. Characterizing finite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59:115–131, 1988.
C. Daws and S. Yovine. Two examples of verification of multirate timed automata with Kronos. In Proceedings of the 16th Annual Real-time Systems Symposium, pages 66–75. IEEE Computer Society Press, 1995.
R. Graham, D. Knuth, and O. Patashnik. Concrete Mathematics. Addison-Wesley Publishing Company, 1989.
T.A. Henzinger. Hybrid automata with finite bisimulations. In Z. Fülöp and F. Gécseg, editors, ICALP 95: Automata, Languages, and Programming, Lecture Notes in Computer Science 944, pages 324–335. Springer-Verlag, 1995.
T.A. Henzinger. The theory of hybrid automata. In Proceedings of the Eleventh Annual Symposium on Logic in Computer Science. IEEE Computer Society Press, 1996.
T.A. Henzinger and P.-H. Ho. Algorithmic analysis of nonlinear hybrid systems. In P. Wolper, editor, CAV 95: Computer-aided Verification, Lecture Notes in Computer Science 939, pages 225–238. Springer-Verlag, 1995.
M.R. Henzinger, T.A. Henzinger, and P.W. Kopke. Computing simulations on finite and infinite graphs. In Proceedings of the 36rd Annual Symposium on Foundations of Computer Science, pages 453–462. IEEE Computer Society Press, 1995.
T.A. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: the next generation. In Proceedings of the 16th Annual Real-time Systems Symposium, pages 56–65. IEEE Computer Society Press, 1995.
T.A. Henzinger and P.W. Kopke. State equivalences for rectangular hybrid automata. Technical Report CSD-TR-96-1588, Cornell University, 1996.
T.A. Henzinger, P.W. Kopke, A. Puri, and P. Varaiya. What's decidable about hybrid automata? In Proceedings of the 27th Annual Symposium on Theory of Computing, pages 373–382. ACM Press, 1995.
P.-H. Ho and H. Wong-Toi. Automated analysis of an audio control protocol. In P. Wolper, editor, CAV 95: Computer-aided Verification, Lecture Notes in Computer Science 939, pages 381–394. Springer-Verlag, 1995.
K.G. Larsen, P. Pettersson, and W. Yi. Compositional and symbolic model checking of real-time systems. In Proceedings of the 16th Annual Real-time Systems Symposium, pages 76–87. IEEE Computer Society Press, 1995.
X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. An approach to the description and analysis of hybrid systems. In R.L. Grossman, A. Nerode, A.P. Ravn, and H. Rischel, editors, Hybrid Systems I, Lecture Notes in Computer Science 736, pages 149–178. Springer-Verlag, 1993.
A. Puri and P. Varaiya. Decidability of hybrid systems with rectangular differential inclusions. In D.L. Dill, editor, CAV 94: Computer-aided Verification, Lecture Notes in Computer Science 818, pages 95–104. Springer-Verlag, 1994.
S. Tripakis and S. Yovine. Analysis of timed systems based on time-abstracting bisimulations. In CAV 96: Computer-aided Verification, Lecture Notes in Computer Science. Springer-Verlag, 1996.
Editors and Affiliations
Rights and permissions
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Henzinger, T.A., Kopke, P.W. (1996). State equivalences for rectangular hybrid automata. In: Montanari, U., Sassone, V. (eds) CONCUR '96: Concurrency Theory. CONCUR 1996. Lecture Notes in Computer Science, vol 1119. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61604-7_74
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61604-7
Online ISBN: 978-3-540-70625-0
eBook Packages: Springer Book Archive