Formal Methods in System Design

, Volume 44, Issue 1, pp 71–90 | Cite as

Safety verification of non-linear hybrid systems is quasi-decidable

  • Stefan Ratschan


Safety verification of hybrid systems is undecidable, except for very special cases. In this paper, we circumvent undecidability by providing a verification algorithm that provably terminates for all robust problem instances, but need not necessarily terminate for non-robust problem instances. A problem instance x is robust iff the given property holds not only for x itself, but also when x is perturbed a little bit. Since, in practice, well-designed hybrid systems are usually robust, this implies that the algorithm terminates for the cases occurring in practice. In contrast to earlier work, our result holds for a very general class of hybrid systems, and it uses a continuous time model.


Hybrid systems Safety verification Decidability Robustness 


  1. 1.
    Asarin E, Bouajjani A (2001) Perturbed Turing machines and hybrid systems. In: Proc LICS’01, pp 269–278 Google Scholar
  2. 2.
    Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhäuser, Boston zbMATHGoogle Scholar
  3. 3.
    Bhatia A, Frazzoli E (2007) Sampling-based resolution-complete safety falsification of linear hybrid systems. In: 46th IEEE conference on decision and control, pp 3405–3411 Google Scholar
  4. 4.
    Bournez O, Campagnolo ML (2008) A survey on continuous time computations. In: Cooper S, Löwe B, Sorbi A (eds) New computational paradigms. Springer, New York, pp 383–423 CrossRefGoogle Scholar
  5. 5.
    Caviness BF, Johnson JR (eds) (1998) Quantifier elimination and cylindrical algebraic decomposition. Springer, Berlin zbMATHGoogle Scholar
  6. 6.
    Cheng P, Kumar V (2008) Sampling-based falsification and verification of controllers for continuous dynamic systems. Int J Robot Res 27(11–12):1232–1245 CrossRefGoogle Scholar
  7. 7.
    Collins P (2005) Continuity and computability of reachable sets. Theor Comput Sci 341:162–195 CrossRefzbMATHGoogle Scholar
  8. 8.
    Collins P (2011) Semantics and computability of the evolution of hybrid systems. SIAM J Control Optim 49(2):890–925 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Damm W, Pinto G, Ratschan S (2007) Guaranteed termination in the verification of LTL properties of non-linear robust discrete time hybrid systems. Int J Found Comput Sci 18(1):63–86 CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Fränzle M (1999) Analysis of hybrid systems: an ounce of realism can save an infinity of states. In: Flum J, Rodriguez-Artalejo M (eds) Computer science logic (CSL’99). LNCS, vol 1683. Springer, Berlin Google Scholar
  11. 11.
    Fränzle M (2001) What will be eventually true of polynomial hybrid automata. In: Kobayashi N, Pierce BC (eds) Theoretical aspects of computer software (TACS 2001). LNCS, vol 2215. Springer, Berlin Google Scholar
  12. 12.
    Goebel R, Sanfelice RG, Teel AR (2012) Hybrid dynamical systems: modeling, stability, and robustness. Princeton University Press, Princeton Google Scholar
  13. 13.
    Goebel R, Teel A (2006) Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica 42(4):573–587 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Henzinger TA, Ho P-H, Wong-Toi H (1998) Algorithmic analysis of nonlinear hybrid systems. IEEE Trans Autom Control 43:540–554 CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Henzinger TA, Kopke PW, Puri A, Varaiya P (1998) What’s decidable about hybrid automata. J Comput Syst Sci 57:94–124 CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Henzinger TA, Raskin J-F (2000) Robust undecidability of timed and hybrid systems. In: Lynch N, Krogh B (eds) Proc HSCC’00. LNCS, vol 1790. Springer, Berlin Google Scholar
  17. 17.
    Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, New York zbMATHGoogle Scholar
  18. 18.
    Moore RE (1966) Interval analysis. Prentice Hall, Englewood Cliffs zbMATHGoogle Scholar
  19. 19.
    Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  20. 20.
    Puri A (2000) Dynamical properties of timed automata. Discrete Event Dyn Syst 10(1):87–113 CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Puri A, Borkar V, Varaiya P (1996) ε-Approximation of differential inclusions. In: Alur R, Henzinger TA, Sontag ED (eds) Hybrid systems. LNCS, vol 1066. Springer, Berlin Google Scholar
  22. 22.
    Ratschan S (2002) Quantified constraints under perturbations. J Symb Comput 33(4):493–505 CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ratschan S (2010) Safety verification of non-linear hybrid systems is quasi-semidecidable. In: TAMC 2010: 7th annual conference on theory and applications of models of computation. LNCS, vol 6108. Springer, Berlin, pp 397–408 Google Scholar
  24. 24.
    Richardson D (1968) Some undecidable problems involving elementary functions of a real variable. J Symb Log 33:514–520 zbMATHGoogle Scholar
  25. 25.
    Swaminathan M, Fränzle M, Katoen J-P (2008) The surprising robustness of (closed) timed automata against clock-drift. In: 5th Ifip int conf on theoretical comp sc, pp 537–553 Google Scholar
  26. 26.
    Tarski A (1951) A decision method for elementary algebra and geometry. University of California Press, Berkeley. Also in [5] zbMATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPragueCzech Republic

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