Series of Abstractions for Hybrid Automata

  • Ashish Tiwari
  • Gaurav Khanna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2289)


We present a technique based on the use of the quantifier elimination decision procedure for real closed fields and simple theorem proving to construct a series of successively finer qualitative abstractions of hybrid automata. The resulting abstractions are always discrete transition systems which can then be used by any traditional analysis tool. The constructed abstractions are conservative and can be used to establish safety properties of the original system. Our technique works on linear and non-linear polynomial hybrid systems, that is, the guards on discrete transitions and the continuous flows in all modes can be specified using arbitrary polynomial expressions over the continuous variables. We have a prototype tool in the SAL environment [13] which is built over the theorem prover PVS [19]. The technique promises to scale well to large and complex hybrid systems.


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  1. [1]
    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 Hybrid Systems [7], pages 209–229.Google Scholar
  2. [2]
    R. Alur, R. Grosu, Y. Hur, V. Kumar, and I. Lee. Modular specifications of hybrid systems in CHARON. In Proc of 3rd Intl Workshop on Hybrid Systems: Computation and Control, volume 1790 of LNCS, pages 6–19, 2000.CrossRefGoogle Scholar
  3. [3]
    R. Alur, T. A Henzinger, G. Lafferriere, and G. J. Pappas. Discrete abstractions of hybrid systems. Proceedings of the IEEE, 88(2):971–984, July 2000.Google Scholar
  4. [4]
    B. Buchberger, G. E. Collins, M. J. Encarnacion, H. Hong, J. R. Johnson, W. Krandick, R. Loos, A. M. Mandache, A. Neubacher, and H. Vielhaber. SACLIB 1.1 user’s guide. In RISC-Linz Report Series, Tech Report No 93–19. Kurt Gödel Institute, 1993.
  5. [5]
    A. Chutinam and B. H. Krogh. Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In Hybrid Systems: Computation and Control, volume 1569 of LNCS, pages 76–90. Springer-Verlag, 1999.CrossRefGoogle Scholar
  6. [6]
    G. E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In Proc. Second GI Conf. AutomataTheory and Formal Languages, volume 33 of LNCS, pages 134–183, 1975.Google Scholar
  7. [7]
    R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel (eds.). Hybrid Systems, volume 736 of LNCS. Springer-Verlag, Berlin, 1993.Google Scholar
  8. [8]
    M. R. Henzinger, T. A. Henzinger, and P. W. Kopke. Computing simulations on finite and infinite graphs. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science FOCS, pages 453–462, 1995.Google Scholar
  9. [9]
    T. A. Henzinger. Hybrid automata with finite bisimulations. In Proc. 22nd ICALP, volume 944 of LNCS, pages 324–335. Springer-Verlag, 1995.Google Scholar
  10. [10]
    T. A. Henzinger, P-H. Ho, and H. Wong-Toi. Algorithmic analysis of nonlinear hybrid systems. IEEE Transactions on Automatic Control, 43:540–554, 1998.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    T. A. Henzinger, P. W. Kopke, A. Puri, and P. Varaiya. What’s decidable about hybrid automata? Journal of Computer and System Sciences, 57:94–124, 1998.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    H. Hong. Quantifier elimination in elementary algebra and geometry by partial cylindrical algebraic decomposition version 13. In The world wide web, 1995.
  13. [14]
    T. Loeser, Y. Iwasaki, and R. Fikes. Safety veri.cation proofs for physical systems. In Proc. of the 12th Intl. Workshop on Qualitative Reasoning, pages 88–95. AAAI Press, 1998.Google Scholar
  14. [15]
    I. Mitchell, A. Bayen, and C. Tomlin. Validating a hamilton-jacobi approximation to hybrid system reachable sets. In M. D. Benedetto and A. L. Sangiovanni-Vincentelli, editors, HSCC 4th Intl. Workshop, volume 2034 of LNCS, 2001.Google Scholar
  15. [16]
    P. J. Mosterman and G. Biswas. Monitoring, prediction, and fault isolation in dynamic physical systems. AAAI-97, pages 100–105, 1997.Google Scholar
  16. [17]
    X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. An approach to the description and analysis of hybrid systems. In Hybrid Systems [7], pages 149–178.Google Scholar
  17. [18]
    A. Olivero, J. Sifakis, and S. Yovine. Using abstractions for the verification of linear hybrid systems. In Proc. of the 6th Computer-Aided Verification CAV, volume 818 of LNCS, pages 81–94, 1994.Google Scholar
  18. [19]
    N. Shankar, S. Owre, and J. M. Rushby. The PVS Proof Checker: A Reference Manual. Computer Science Lab, SRI International, 1993.Google Scholar
  19. [20]
    B. Shults and B. J. Kuipers. Proving properties of continuous systems: qualitative simulation and temporal logic. AI Journal, 92:91–129, 1997.MATHMathSciNetGoogle Scholar
  20. [21]
    O. Sokolsky and H. S. Hong. Qualitative modeling of hybrid systems. In Proc. of the Montreal Workshop, 2001. Available from
  21. [22]
    O. Stursberg, S. Kowalewski, I. Hoffmann, and Preußig. Comparing timed and hybrid automata as approximations of continuous systems. In P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, editors, Hybrid Systems IV, volume 1273 of LNCS, pages 361–377. Springer-Verlag, 1997.CrossRefGoogle Scholar
  22. [23]
    A. Tarski. A Decision Method for Elementary Algebra and Geometry. University of California Press, 1948. Second edition.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ashish Tiwari
    • 1
  • Gaurav Khanna
    • 2
  1. 1.SRI InternationalMenlo ParkUSA
  2. 2.Theoretical and Computational Studies GroupLong Island University, SouthamptonSouthampton

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