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The Büchi Complementation Saga

  • Moshe Y. Vardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems are reduced, involves complementation. For automata on finite words, which correspond to safety properties, complementation involves determinization. The 2 n blow-up that is caused by the subset construction is justified by a tight lower bound. For Büchi automata on infinite words, which are required for the modeling of liveness properties, optimal complementation constructions are quite complicated, as the subset construction is not sufficient. We review here progress on this problem, which dates back to its introduction in Büchi’s seminal 1962 paper.

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References

  1. 1.
    Armoni, R., et al.: The ForSpec temporal logic: A new temporal property-specification logic. In: Katoen, J.-P., Stevens, P. (eds.) ETAPS 2002 and TACAS 2002. LNCS, vol. 2280, pp. 296–311. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Birget, J.C.: Partial orders on words, minimal elements of regular languages, and state complexity. Theoretical Computer Science 119, 267–291 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proc. International Congress on Logic, Method, and Philosophy of Science, 1960, pp. 1–12. Stanford University Press, Stanford (1962)Google Scholar
  4. 4.
    Burch, J.R., et al.: Symbolic model checking: 1020 states and beyond. Information and Computation 98(2), 142–170 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Daniele, N., Guinchiglia, F., Vardi, M.Y.: Improved automata generation for linear temporal logic. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 249–260. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Emerson, E.A., Clarke, E.M.: Characterizing correctness properties of parallel programs using fixpoints. In: Proc. 7th InternationalColloq. on Automata, Languages and Programming, pp. 169–181 (1980)Google Scholar
  7. 7.
    Emerson, E.A., Lei, C.-L.: Temporal model checking under generalized fairness constraints. In: Proc. 18th Hawaii International Conference on System Sciences, Western Periodicals Company, North Holywood (1985)Google Scholar
  8. 8.
    Fisler, K., et al.: Is there a best symbolic cycle-detection algorithm? In: Margaria, T., Yi, W. (eds.) ETAPS 2001 and TACAS 2001. LNCS, vol. 2031, pp. 420–434. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Friedgut, E., Kupferman, O., Vardi, M.Y.: Büchi complementation made tighter. Int’l J. of Foundations of Computer Science 17(4), 851–867 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Gastin, P., Oddoux, D.: Fast LTL to büchi automata translation. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 53–65. Springer, Heidelberg (2001)Google Scholar
  11. 11.
    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
  12. 12.
    Gurumurthy, S., et al.: On complementing nondeterministic Büchi automata. In: Geist, D., Tronci, E. (eds.) CHARME 2003. LNCS, vol. 2860, pp. 96–110. Springer, Heidelberg (2003)Google Scholar
  13. 13.
    Henzinger, T.A., Kupferman, O., Rajamani, S.: Fair simulation. Information and Computation 173(1), 64–81 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Holzmann, G.J.: The model checker SPIN. IEEE Trans. on Software Engineering, Special issue on Formal Methods in Software Practice 23(5), 279–295 (1997)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kesten, Y., Piterman, N., Pnueli, A.: Bridging the gap between fair simulation and trace containment. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 381–393. Springer, Heidelberg (2003)Google Scholar
  16. 16.
    Klarlund, N.: Progress measures for complementation of ω-automata with applications to temporal logic. In: Proc. 32nd IEEE Symp. on Foundations of Computer Science, San Juan, October 1991, pp. 358–367. IEEE Computer Society Press, Los Alamitos (1991)CrossRefGoogle Scholar
  17. 17.
    Kupferman, O., Vardi, M.Y.: Weak alternating automata are not that weak. In: Proc. 5th Israeli Symp. on Theory of Computing and Systems, pp. 147–158. IEEE Computer Society Press, Los Alamitos (1997)CrossRefGoogle Scholar
  18. 18.
    Kupferman, O., Vardi, M.Y.: Weak alternating automata are not that weak. ACM Trans. on Computational Logic 2(2), 408–429 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Kupferman, O., Vardi, M.Y.: Complementation constructions for nondeterministic automata on infinite words. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 206–221. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Kupferman, O., Vardi, M.Y.: From complementation to certification. Theoretical Computer Science 305, 591–606 (2005)MathSciNetGoogle Scholar
  21. 21.
    Kurshan, R.P.: Computer Aided Verification of Coordinating Processes. Princeton University Press, Princeton (1994)Google Scholar
  22. 22.
    Löding, C.: Optimal bounds for the transformation of omega-automata. In: Pandu Rangan, C., Raman, V., Ramanujam, R. (eds.) FST TCS 1999. LNCS, vol. 1738, pp. 97–109. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  23. 23.
    Michel, M.: Complementation is more difficult with automata on infinite words. In: CNET, Paris (1988)Google Scholar
  24. 24.
    Miyano, S., Hayashi, T.: Alternating finite automata on ω-words. Theoretical Computer Science 32, 321–330 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Muller, D.E.: Infinite sequences and finite machines. In: Proc. 4th IEEE Symp. on Switching Circuit Theory and Logical design, pp. 3–16. IEEE Computer Society Press, Los Alamitos (1963)Google Scholar
  26. 26.
    Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. Theoretical Computer Science 54, 267–276 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Pécuchet, J.P.: On the complementation of büchi automata. Theor. Comput. Sci. 47(3), 95–98 (1986)CrossRefzbMATHGoogle Scholar
  28. 28.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3, 115–125 (1959)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Safra, S.: On the complexity of ω-automata. In: Proc. 29th IEEE Symp. on Foundations of Computer Science, White Plains, October 1988, pp. 319–327. IEEE Computer Society Press, Los Alamitos (1988)Google Scholar
  30. 30.
    Safra, S.: Exponential determinization for ω-automata with strong-fairness acceptance condition. In: Proc. 24th ACM Symp. on Theory of Computing, Victoria, May 1992, ACM Press, New York (1992)Google Scholar
  31. 31.
    Sakoda, W., Sipser, M.: Non-determinism and the size of two-way automata. In: Proc. 10th ACM Symp. on Theory of Computing, pp. 275–286. ACM Press, New York (1978)Google Scholar
  32. 32.
    Sistla, A.P., Vardi, M.Y., Wolper, P.: The complementation problem for Büchi automata with applications to temporal logic. Theoretical Computer Science 49, 217–237 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    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
  34. 34.
    Temme, N.M.: Asimptotic estimates of Stirling numbers. Stud. Appl. Math. 89, 233–243 (1993)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Thomas, W.: Complementation of Büchi automata revised. In: Karhumäki, J., et al. (eds.) Jewels are Forever, pp. 109–120. Springer, Heidelberg (1999)Google Scholar
  36. 36.
    Vardi, M.Y.: Verification of concurrent programs - the automata-theoretic framework. Annals of Pure and Applied Logic 51, 79–98 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Wolper, P.: Temporal logic can be more expressive. Information and Control 56(1–2), 72–99 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    Yan, Q.: Lower bounds for complementation of ω-automata via the full automata technique. In: Bugliesi, M., et al. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 589–600. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Moshe Y. Vardi
    • 1
  1. 1.Rice University, Department of Computer Science, Rice University, Houston, TX 77251-1892U.S.A.

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