Combining Symbolic Representations for Solving Timed Games

  • Rüdiger Ehlers
  • Robert Mattmüller
  • Hans-Jörg Peter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6246)

Abstract

We present a general approach to combine symbolic state space representations for the discrete and continuous parts in the synthesis of winning strategies for timed reachability games. The combination is based on abstraction refinement where discrete symbolic techniques are used to produce a sequence of abstract timed game automata. After each refinement step, the resulting abstraction is used for computing an under- and an over-approximation of the timed winning states. The key idea is to identify large relevant and irrelevant parts of the precise weakest winning strategy already on coarse, and therefore simple, abstractions. If neither the existence nor nonexistence of a winning strategy can be established in the approximations, we use them to guide the refinement process. Based on a prototype that combines binary decision diagrams[7,9] and difference bound matrices[5], we experimentally evaluate the technique on standard benchmarks from timed controller synthesis. The results clearly demonstrate the potential of the new approach concerning running time and memory consumption compared to the classical on-the-fly algorithm implemented in Uppaal-Tiga [10,4].

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References

  1. 1.
    Altisen, K., Tripakis, S.: Tools for controller synthesis of timed systems. In: 2nd Workshop on Real-Time Tools, RT-TOOLS (2002)Google Scholar
  2. 2.
    Alur, R., Dill, D.L.: A theory of timed automata. Theo. Comp. Sci. 126(2) (1994)Google Scholar
  3. 3.
    Asarin, E., Maler, O., Pnueli, A., Sifakis, J.: Controller synthesis for timed automata. In: Proc. 5th IFAC Conference on System Structure and Control (1998)Google Scholar
  4. 4.
    Behrmann, G., Cougnard, A., David, A., Fleury, E., Larsen, K.G., Lime, D.: UPPAAL-Tiga: Time for playing games! In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 121–125. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Bengtsson, J.: Clocks, DBM, and States in Timed Systems. PhD thesis, Uppsala University (2002)Google Scholar
  6. 6.
    Brückner, I., Dräger, K., Finkbeiner, B., Wehrheim, H.: Slicing abstractions. In: vol. 89, pp. 369–392. IOS Press, Amsterdam (2008)Google Scholar
  7. 7.
    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Computers 35(8), 677–691 (1986)MATHCrossRefGoogle Scholar
  8. 8.
    Bulychev, P., Chatain, T., David, A., Larsen, K.G.: Efficient on-the-fly algorithm for checking alternating timed simulation. In: Ouaknine, J., Vaandrager, F.W. (eds.) FORMATS 2009. LNCS, vol. 5813, pp. 73–87. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. Inf. Comput. 98(2) (1992)Google Scholar
  10. 10.
    Cassez, F., David, A., Fleury, E., Larsen, K.G., Lime, D.: Efficient on-the-fly algorithms for the analysis of timed games. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 66–80. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    de Alfaro, L., Roy, P.: Solving games via three-valued abstraction refinement. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 74–89. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Henzinger, T.A., Jhala, R., Majumdar, R.: Counterexample-guided control. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Henzinger, T.A., Kopke, P.W.: Discrete-time control for rectangular hybrid automata. Theoretical Computer Science 221(1-2), 369–392 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lewerentz, C., Lindner, T. (eds.): Formal Development of Reactive Systems - Case Study Production Cell (1995)Google Scholar
  15. 15.
    Maler, O., Pnueli, A., Sifakis, J.: On the synthesis of discrete controllers for timed systems (an extended abstract). In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 229–242. Springer, Heidelberg (1995)Google Scholar
  16. 16.
    Peter, H.J., Mattmüller, R.: Component-based abstraction refinement for timed controller synthesis. In: RTSS (2009)Google Scholar
  17. 17.
    Sentovich, E., Singh, K., Lavagno, L., Moon, C., Murgai, R., Saldanha, A., Savoj, H., Stephan, P., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: SIS: A system for sequential circuit synthesis. Technical report, University of California (1992)Google Scholar
  18. 18.
    Somenzi, F.: CUDD: CU Decision Diagram package release 2.4.2 (2009)Google Scholar
  19. 19.
    Thomas, W.: On the synthesis of strategies in infinite games. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, Springer, Heidelberg (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Rüdiger Ehlers
    • 1
  • Robert Mattmüller
    • 2
  • Hans-Jörg Peter
    • 1
  1. 1.Reactive Systems GroupSaarland UniversityGermany
  2. 2.Foundations of Artificial Intelligence GroupFreiburg UniversityGermany

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