Advertisement

Simulation Relations for Alternating Parity Automata and Parity Games

  • Carsten Fritz
  • Thomas Wilke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4036)

Abstract

We adapt the notion of delayed simulation to alternating parity automata and parity games. On the positive side, we show that (i) the corresponding simulation relation can be computed in polynomial time and (ii) delayed simulation implies language inclusion. On the negative side, we point out that quotienting with respect to delayed simulation does not preserve the language recognized, which means that delayed simulation cannot be used for state-space reduction via merging of simulation equivalent states. As a remedy, we introduce finer, so-called biased notions of delayed simulation where we show quotienting does preserve the language recognized. We propose a heuristic for reducing the size of alternating parity automata and parity games and, as an evidence for its usefulness, demonstrate that it is successful when applied to the Jurdziński family of parity games.

Keywords

Existential State Linear Temporal Logic Winning Strategy Universal State Tree Automaton 
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.
    Milner, R.: An algebraic definition of simulation between programs. In: Cooper, D.C. (ed.) Proc. 2nd Internat. Joint Conf. on Artificial Intelligence, London, UK, pp. 481–489. William Kaufmann, San Francisco (1971)Google Scholar
  2. 2.
    Etessami, K., Holzmann, G.: Optimizing Büchi automata. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 153–167. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Etessami, K.: (Temporal massage parlor), available at: http://www.bell-labs.com/project/TMP/
  4. 4.
    Somenzi, F., Bloem, R.: Efficient Büchi automata from LTL formulae. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 248–263. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Gurumurthy, S., Bloem, R., Somenzi, F.: Fair simulation minimization. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 610–623. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Bloem, R.: Wring: an LTL to Buechi translator, available at: http://www.ist.tugraz.at/staff/bloem/wring.html
  7. 7.
    Fritz, C.: Constructing Büchi automata from linear temporal logic using simulation relations for alternating Büchi automata. In: Ibarra, O.H., Dang, Z. (eds.) CIAA 2003. LNCS, vol. 2759, pp. 35–48. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Fritz, C., Teegen, B.: LTL → NBA (improved version), available at: http://www.ti.informatik.uni-kiel.de/~fritz/ABA-Simulation/ltl.cgi
  9. 9.
    Fritz, C.: Concepts of automata construction from LTL. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS, vol. 3835, pp. 728–742. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Mostowski, A.W.: Regular expressions for infinite trees and a standard form of automata. In: Skowron, A. (ed.) SCT 1984. LNCS, vol. 208, pp. 157–168. Springer, Heidelberg (1985)Google Scholar
  11. 11.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy (extended abstract). In: Proc. 32nd Ann. Symp. on Foundations of Computer Science (FoCS 1991), San Juan, Puerto Rico, pp. 368–377. IEEE Computer Society Press, Los Alamitos (1991)CrossRefGoogle Scholar
  12. 12.
    Etessami, K., Wilke, T., Schuller, R.A.: Fair simulation relations, parity games, and state space reduction for büchi automata. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 694–707. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Etessami, K., Wilke, T., Schuller, R.A.: Fair simulation relations, parity games, and state space reduction for Büchi automata. SIAM J. Comput. 34(5), 1159–1175 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Henzinger, T.A., Rajamani, S.K.: Fair bisimulation. In: Schwartzbach, M.I., Graf, S. (eds.) TACAS 2000. LNCS, vol. 1785, pp. 299–314. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  16. 16.
    Fritz, C.: Simulation-Based Simplification of omega-Automata. PhD thesis, Technische Fakultät der Christian-Albrechts-Universität zu Kiel (2005), available at: http://e-diss.uni-kiel.de/diss_1644/
  17. 17.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research (outcome of a Dagstuhl seminar, February 2001). In: Grädel, E., Thomas, W., Wilke, T. (eds.) Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)Google Scholar
  18. 18.
    Gurevich, Y., Harrington, L.: Trees, automata, and games. In: 14th ACM Symp. on the Theory of Computing, pp. 60–65. ACM Press, San Francisco (1982)Google Scholar
  19. 19.
    V’öge, J., Jurdziński, M.: A discrete strategy improvement algorithm for solving parity games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Fritz, C., Wilke, T.: Simulation relations for alternating Büchi automata. Theoretical Computer Science 338(1–3), 275–314 (2005)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Calude, C., Calude, E., Khoussainov, B.: Finite nondeterministic automata: Simulation and minimality. Theor. Comput. Sci. 242(1-2), 219–235 (2000)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Carsten Fritz
    • 1
  • Thomas Wilke
    • 1
  1. 1.Christian-Albrechts-Universität zu Kiel 

Personalised recommendations