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Annual Symposium on Theoretical Aspects of Computer Science

STACS 2007: STACS 2007 pp 12–22Cite as

The Büchi Complementation Saga

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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 2n 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. 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)

    Chapter  Google Scholar 

  2. Birget, J.C.: Partial orders on words, minimal elements of regular languages, and state complexity. Theoretical Computer Science 119, 267–291 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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. Burch, J.R., et al.: Symbolic model checking: 1020 states and beyond. Information and Computation 98(2), 142–170 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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. Henzinger, T.A., Kupferman, O., Rajamani, S.: Fair simulation. Information and Computation 173(1), 64–81 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  18. Kupferman, O., Vardi, M.Y.: Weak alternating automata are not that weak. ACM Trans. on Computational Logic 2(2), 408–429 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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. Kupferman, O., Vardi, M.Y.: From complementation to certification. Theoretical Computer Science 305, 591–606 (2005)

    MathSciNet  Google Scholar 

  21. Kurshan, R.P.: Computer Aided Verification of Coordinating Processes. Princeton University Press, Princeton (1994)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  23. Michel, M.: Complementation is more difficult with automata on infinite words. In: CNET, Paris (1988)

    Google Scholar 

  24. Miyano, S., Hayashi, T.: Alternating finite automata on ω-words. Theoretical Computer Science 32, 321–330 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  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. Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. Theoretical Computer Science 54, 267–276 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pécuchet, J.P.: On the complementation of büchi automata. Theor. Comput. Sci. 47(3), 95–98 (1986)

    Article  MATH  Google Scholar 

  28. Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3, 115–125 (1959)

    Article  MathSciNet  Google Scholar 

  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. 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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  34. Temme, N.M.: Asimptotic estimates of Stirling numbers. Stud. Appl. Math. 89, 233–243 (1993)

    MathSciNet  MATH  Google Scholar 

  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. Vardi, M.Y.: Verification of concurrent programs - the automata-theoretic framework. Annals of Pure and Applied Logic 51, 79–98 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  37. Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wolper, P.: Temporal logic can be more expressive. Information and Control 56(1–2), 72–99 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

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Authors and Affiliations

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

    Moshe Y. Vardi

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  1. Moshe Y. Vardi
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Wolfgang Thomas Pascal Weil

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Vardi, M.Y. (2007). The Büchi Complementation Saga. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_2

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  • DOI: https://doi.org/10.1007/978-3-540-70918-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70917-6

  • Online ISBN: 978-3-540-70918-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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