The Image Computation Problem in Hybrid Systems Model Checking

  • André Platzer
  • Edmund M. Clarke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4416)


In this paper, we analyze limits of approximation techniques for (non-linear) continuous image computation in model checking hybrid systems. In particular, we show that even a single step of continuous image computation is not semidecidable numerically even for a very restricted class of functions. Moreover, we show that symbolic insight about derivative bounds provides sufficient additional information for approximation refinement model checking. Finally, we prove that purely numerical algorithms can perform continuous image computation with arbitrarily high probability. Using these results, we analyze the prerequisites for a safe operation of the roundabout maneuver in air traffic collision avoidance.


model checking hybrid systems image computation 


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  1. 1.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  2. 2.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)zbMATHGoogle Scholar
  3. 3.
    Fränzle, M.: Analysis of hybrid systems. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 126–140. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 137–151. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Anai, H., Weispfenning, V.: Reach set computations using real quantifier elimination. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 63–76. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Piazza, C., et al.: Algorithmic algebraic model checking I: Challenges from systems biology. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, Springer, Heidelberg (2005)Google Scholar
  7. 7.
    Blum, L., et al.: Complexity and real computation. Springer, New York (1998)Google Scholar
  8. 8.
    Mora, T.: Solving Polynomial Equation Systems II. Cambridge Univ. Press, Cambridge (2005)zbMATHGoogle Scholar
  9. 9.
    Tomlin, C., Pappas, G.J., Sastry, S.: Conflict resolution for air traffic management. IEEE Transactions on Automatic Control 43(4), 509–521 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Massink, M., Francesco, N.D.: Modelling free flight with collision avoidance. In: ICECCS, pp. 270–280. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  11. 11.
    Alur, R., Henzinger, T.A., Ho, P.-H.: Automatic symbolic verification of embedded systems. IEEE Trans. Software Eng. 22(3), 181–201 (1996)CrossRefGoogle Scholar
  12. 12.
    Silva, B.I., et al.: Modeling and verification of hybrid dynamical system using CheckMate. In: ADPM (2000)Google Scholar
  13. 13.
    Frehse, G.: PHAVer: Algorithmic verification of hybrid systems past HyTech. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Damm, W., Pinto, G., Ratschan, S.: Guaranteed termination in the verification of LTL properties of non-linear robust discrete time hybrid systems. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Lanotte, R., Tini, S.: Taylor approximation for hybrid systems. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 402–416. Springer, Heidelberg (2005)Google Scholar
  16. 16.
    Clarke, E.M., et al.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Stone, M.H.: The generalised Weierstrass approximation theorem. Math. Mag. 21, 167–184 and 237–254 (1948)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bejancu, A.: The uniform convergence of multivariate natural splines. Technical Report NA1997/07, Applied Mathematics, Cambridge, UK (1997)Google Scholar
  19. 19.
    Wang, R.-H.: Multivariate Spline Functions and Their Applications. Kluwer Academic Publishers, Dordrecht (2001)zbMATHGoogle Scholar
  20. 20.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (2002)zbMATHGoogle Scholar
  21. 21.
    Asarin, E., Dang, T., Girard, A.: Reachability analysis of nonlinear systems using conservative approximation. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 20–35. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. 22.
    Morari, M., Thiele, L. (eds.): HSCC 2005. LNCS, vol. 3414. Springer, Heidelberg (2005)zbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • André Platzer
    • 1
  • Edmund M. Clarke
    • 2
  1. 1.University of Oldenburg, Department of Computing ScienceGermany
  2. 2.Carnegie Mellon University, Computer Science Department, Pittsburgh, PA 

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