Robust Undecidability of Timed and Hybrid Systems

  • Thomas A. Henzinger
  • Jean-François Raskin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1790)


The algorithmic approach to the analysis of timed and hybrid systems is fundamentally limited by undecidability, of universality in the timed case (where all continuous variables are clocks), and of emptiness in the rectangular case (which includes drifting clocks). Traditional proofs of undecidability encode a single Turing computation by a single timed trajectory. These proofs have nurtured the hope that the introduction of “fuzziness” into timed and hybrid models (in the sense that a system cannot distinguish between trajectories that are sufficiently similar) may lead to decidability. We show that this is not the case, by sharpening both fundamental undecidability results. Besides the obvious blow our results deal to the algorithmic method, they also prove that the standard model of timed and hybrid systems, while not “robust” in its definition of trajectory acceptance (which is affected by tiny perturbations in the timing of events), is quite robust in its mathematical properties: the undecidability barriers are not affected by reasonable perturbations of the model.


Hybrid System Positive Real Hybrid Automaton Time Automaton Open Slot 
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  1. [AD94]
    R. Alur and D.L. Dill. A theory of timed automata. Theoretical Computer Science, 126:183–235, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [AFH94]
    R. Alur, L. Fix, and T.A. Henzinger. A determinizable class of timed automata, CAV 94: Computer-aided Verification, LNCS 818, Springer Verlag, 1–13, 1994.Google Scholar
  3. [AFH96]
    R. Alur, T. Feder, and T.A. Henzinger. The benefits of relaxing punctuality. Journal of the ACM, 43(1):116–146, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [AMP95]
    E. Asarin, O. Maler, and A. Pnueli. Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science, 138:65–66, 1995.CrossRefMathSciNetGoogle Scholar
  5. [BT99]
    V. D. Blondel and J. N. Tsitsiklis. A survey of computational complexity results in systems and control. To appear in Automatica, 1999.Google Scholar
  6. [Fra99]
    M. Franzle. Analysis of Hybrid Systems: An ounce of realism can save an infinity of states, CSL’99: Computer Science Logic. LNCS 1683, Springer Verlag, 126–140, 1999.CrossRefGoogle Scholar
  7. [GHJ97]
    V. Gupta, T.A. Henzinger, and R. Jagadeesan. Robust timed automata, HART 97: Hybrid and Real-time Systems. LNCS 1201, Springer-Verlag, 331–345, 1997.CrossRefGoogle Scholar
  8. [HK97]
    T.A. Henzinger and P.W. Kopke Discrete-time control for rectangular hybrid automata. ICALP 97: Automata, Languages, and Programming. LNCS 1256, Springer-Verlag, 582–593, 1997.Google Scholar
  9. [HKPV95]
    T.A. Henzinger, P.W. Kopke, A. Puri, and P. Varaiya. What’s decidable about hybrid automata? In 27th Annual Symposium on Theory of Computing, ACM Press, 373–382, 1995.Google Scholar
  10. [HR99]
    T.A. Henzinger and J.-F. Raskin. Robust Undecidability of Timed and Hybrid Systems. Technical Report of the Computer Science Department of the University of California at Berkeley, UCB/CSD-99-1073, October 1999.Google Scholar
  11. [Her98]
    P. Herrmann. Timed automata and recognizability. Information Processing Letters, 65(6):313–318, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [RS99]
    J.-F. Raskin and P.-Y. Schobbens. The Logic of Event Clocks: Decidability, Complexity, and Expressiveness. Journal of Automata, Languages and Combinatorics, 4(3):247–284, 1999.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Thomas A. Henzinger
    • 1
  • Jean-François Raskin
    • 1
    • 2
  1. 1.Department of Electrical Engineering and Computer SciencesUniversity of California at BerkeleyUSA
  2. 2.Département d’Informatique, Faculté des SciencesUniversité Libre de BruxellesBelgium

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