Verification Using Simulation

  • Antoine Girard
  • George J. Pappas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3927)


Verification and simulation have always been complementary, if not competing, approaches to system design. In this paper, we present a novel method for so-called metric transition systems that bridges the gap between verification and simulation, enabling system verification using a finite number of simulations. The existence of metrics on the system state and observation spaces, which is natural for continuous systems, allows us to capitalize on the recently developed framework of approximate bisimulations, and infer the behavior of neighborhood of system trajectories around a simulated trajectory. For nondeterministic linear systems that are robustly safe or robustly unsafe, we provide not only a completeness result but also an upper bound on the number of simulations required as a function of the distance between the reachable set and the unsafe set. Our framework is the first simulation-based verification method that enjoys completeness for infinite-state systems. The complexity is low for robustly safe or robustly unsafe systems, and increases for nonrobust problems. This provides strong evidence that robustness dramatically impacts the complexity of system verification and design.


Hybrid System Transition System Linear Matrix Inequality Transition Relation Linear Temporal Logic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur, R., Grosu, R., Hur, Y., Kumar, V., Lee, I.: Modular specification of hybrid systems in charon. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, p. 6. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Kapinski, J., Krogh, B.H., Maler, O., Stursberg, O.: On systematic simulation of open continuous systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 283–297. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Lee, E.A., Zheng, H.: Operational semantics of hybrid systems. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, p. 25. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Alur, R., Dang, T., Ivancic, F.: Counter-example guided predicate abstraction of hybrid systems. In: Garavel, H., Hatcliff, J. (eds.) ETAPS 2003 and TACAS 2003. LNCS, vol. 2619, pp. 208–223. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Clarke, E., Fehnker, A., Han, Z., Krogh, B., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. International Journal of Foundations of Computer Science 14(4) (2003)Google Scholar
  6. 6.
    Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, p. 477. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Mitchell, I., Tomlin, C.: Level set methods for computation in hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, Springer, Heidelberg (2000)Google Scholar
  8. 8.
    Frehse, G.: Phaver: Algorithmic verification of hybrid systems past hytech. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 258–273. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Girard, A.: Reachability of uncertain linear systems using zonotopes. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 291–305. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    de Alfaro, L., Faella, M., Stoelinga, M.: Linear and branching metrics for quantitative transition systems. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1150–1162. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Girard, A., Pappas, G.J.: Approximation metrics for discrete and continuous systems. Technical Report MS-CIS-05-10, Dept. of CIS, University of Pennsylvania (2005)Google Scholar
  12. 12.
    Girard, A., Pappas, G.J.: Approximate bisimulations for constrained linear systems. In: Proc. IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, pp. 4700–4705 (2005)Google Scholar
  13. 13.
    Girard, A., Pappas, G.J.: Approximate bisimulations for nonlinear dynamical systems. In: Proc. IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, pp. 684–689 (2005)Google Scholar
  14. 14.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (2000)Google Scholar
  15. 15.
    Haghverdi, E., Tabuada, P., Pappas, G.J.: Bisimulation relations for dynamical, control, and hybrid systems. Theoretical Computer Science 342(2-3), 229–262 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pappas, G.J.: Bisimilar linear systems. Automatica 39(12), 2035–2047 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    van der Schaft, A.: Equivalence of dynamical systems by bisimulation. IEEE Transactions on Automatic Control 49(12), 2160–2172 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sturm, J.F.: Using SEDUMI 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Softwares 11-12, 625–653 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Antoine Girard
    • 1
  • George J. Pappas
    • 1
  1. 1.Department of Electrical and Systems EngineeringUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations