Advertisement

A Formalized Theory for Verifying Stability and Convergence of Automata in PVS

  • Sayan Mitra
  • K. Mani Chandy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5170)

Abstract

Correctness of many hybrid and distributed systems require stability and convergence guarantees. Unlike the standard induction principle for verifying invariance, a theory for verifying stability or convergence of automata is currently not available. In this paper, we formalize one such theory proposed by Tsitsiklis [27]. We build on the existing PVS metatheory for untimed, timed, and hybrid input/output automata, and incorporate the concepts about fairness, stability, Lyapunov-like functions, and convergence. The resulting theory provides two sets of sufficient conditions, which when instantiated and verified for particular automata, guarantee convergence and stability, respectively.

Keywords

Distance Function Topological Structure Formalize Theory Mobile Agent Nonempty Subset 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Tempo toolset, version 0.2.2 beta (January 2008), http://www.veromodo.com/tempo/
  2. 2.
    Archer, M.: PVS Strategies for special purpose theorem proving. Annals of Mathematics and Artificial Intelligence 29(1/4) (February 2001)Google Scholar
  3. 3.
    Archer, M., Heitmeyer, C., Sims, S.: TAME: A PVS interface to simplify proofs for automata models. In: Proceedings of UITP 1998 (July 1998)Google Scholar
  4. 4.
    Archer, M., Lim, H., Lynch, N., Mitra, S., Umeno, S.: Specifying and proving properties of timed I/O automata using Tempo. Design Automation for Embedded Systems (to appear, 2008)Google Scholar
  5. 5.
    Bulwahn, L., Krauss, A., Nipkow, T.: Finding lexicographic orders for termination proofs in Isabelle/HOL. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 38–53. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Chandy, K.M., Mitra, S., Pilotto, C.: Formations of mobile agents with message loss and delay (preprint) (2007), http://www.ist.caltech.edu/~mitras/research/2008/asynchcoord.pdf
  7. 7.
    Devillers, M.: Translating IOA automata to PVS. Technical Report CSI-R9903, Computing Science Institute, University of Nijmegen (February 1999), http://www.cs.ru.nl/research/reports/info/CSI-R9903.html
  8. 8.
    Filliâtre, J.: Finite automata theory in Coq: A constructive proof of kleene’s theorem. Technical report, LIP -ENS, Research Report 97-04, Lyon (February 1997)Google Scholar
  9. 9.
    Floyd, R.: Assigning meanings to programs. In: Symposium on Applied Mathematics. Mathematical Aspects of Computer Science, pp. 19–32. American Mathematical Society (1967)Google Scholar
  10. 10.
    Gottliebsen, H.: Transcendental functions and continuity checking in PVS. In: Aagaard, M.D., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 197–214. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Harrison, J.: Theorem Proving with the Real Numbers. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kaynar, D.K., Lynch, N., Segala, R., Vaandrager, F.: The Theory of Timed I/O Automata. Synthesis Lectures on Computer Science. Morgan Claypool, Technical Report MIT-LCS-TR-917 (November 2005)Google Scholar
  13. 13.
    Lester, D.: NASA langley PVS library for topological spaces, http://shemesh.larc.nasa.gov/fm/ftp/larc/PVS-library/topology-details.html
  14. 14.
    Liberzon, D.: Switching in Systems and Control. Systems and Control: Foundations and Applications. Birkhauser, Boston (2003)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lim, H., Kaynar, D., Lynch, N., Mitra, S.: Translating timed I/O automata specifications for theorem proving in PVS. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Luenberger, D.G.: Introduction to Dynamic Systems: Theory, Models, and Applications. John Wiley and Sons, Inc, New York (1979)zbMATHGoogle Scholar
  17. 17.
    Lynch, N., Tuttle, M.: An introduction to Input/Output automata. CWI-Quarterly 2(3), 219–246 (1989)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mitra, S.: A Verification Framework for Hybrid Systems. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA 02139 (September 2007)Google Scholar
  19. 19.
    Mitra, S., Archer, M.: PVS strategies for proving abstraction properties of automata. Electronic Notes in Theoretical Computer Science 125(2), 45–65 (2005)CrossRefzbMATHGoogle Scholar
  20. 20.
    Müller, O.: I/O automata and beyond: Temporal logic and abstraction in Isabelle. In: Proceedings of the 11th International Conference on Theorem Proving in Higher Order Logics, London, UK, pp. 331–348. Springer, London (1998)CrossRefGoogle Scholar
  21. 21.
    Nipkow, T., Slind, K.: I/O automata in Isabelle/HOL. In: Smith, J., Dybjer, P., Nordström, B. (eds.) TYPES 1994. LNCS, vol. 996, pp. 101–119. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  22. 22.
    Müller, O.: A Verification Environment for I/O Automata Based on Formalized Meta-Theory. PhD thesis, Technische Universität München (September 1998)Google Scholar
  23. 23.
    Owre, S., Rajan, S., Rushby, J., Shankar, N., Srivas, M.: PVS: Combining specification, proof checking, and model checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 411–414. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  24. 24.
    Paulin-Mohring, C.: Modelisation of timed automata in Coq. In: Kobayashi, N., Pierce, B.C. (eds.) TACS 2001. LNCS, vol. 2215, pp. 298–315. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  25. 25.
    Paulson, L.C.: Mechanizing UNITY in Isabelle. ACM Transactions on Computational Logic 1(1), 3–32 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rohwedder, E., Pfenning, F.: Mode and termination checking for higher-order logic programs. In: Riis Nielson, H. (ed.) ESOP 1996. LNCS, vol. 1058, pp. 296–310. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  27. 27.
    Tsitsiklis, J.N.: On the stability of asynchronous iterative processes. Theory of Computing Systems 20(1), 137–153 (1987)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Umeno, S., Lynch, N.A.: Safety verification of an aircraft landing protocol: A refinement approach. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 557–572. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sayan Mitra
    • 1
  • K. Mani Chandy
    • 1
  1. 1.California Institute of TechnologyPasadenaUSA

Personalised recommendations