Short Witnesses and Accepting Lassos in ω-Automata

  • Rüdiger Ehlers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6031)


Emptiness checking of ω-automata is a fundamental part of the automata-theoretic toolbox and is frequently applied in many applications, most notably verification of reactive systems. In this particular application, the capability to extract accepted words or alternatively accepting runs in case of non-emptiness is particularly useful, as these have a diagnostic value. However, non-optimised such words or runs can become huge, limiting their usability in practice, thus solutions with a small representation should be preferred. In this paper, we review the known results on obtaining these and complete the complexity landscape for all commonly used automaton types. We also prove upper and lower bounds on the approximation hardness of these problems.


Model Check Acceptance Condition Linear Time Temporal Logic Regular Model Check Approximation Hardness 
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.


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  1. 1.
    Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  2. 2.
    Büchi, J.R.: On a decision method in restricted second-order arithmetic. In: Proc. 1960 Int. Congr. for Logic, Methodology, and Philosophy of Science, pp. 1–11 (1962)Google Scholar
  3. 3.
    Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching-time temporal logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  4. 4.
    Clarke, E.M., Grumberg, O., McMillan, K.L., Zhao, X.: Efficient generation of counterexamples and witnesses in symbolic model checking. In: DAC, pp. 427–432 (1995)Google Scholar
  5. 5.
    Dinur, I., Guruswami, V., Khot, S., Regev, O.: A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput. 34(5), 1129–1146 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Engebretsen, L., Karpinski, M.: Approximation hardness of TSP with bounded metrics. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 201–212. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Farwer, B.: ω-automata. In: [10], pp. 3–20Google Scholar
  8. 8.
    Frieze, A.M., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12(1), 23–39 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gastin, P., Moro, P.: Minimal counterexample generation for SPIN. In: Bošnački, D., Edelkamp, S. (eds.) SPIN 2007. LNCS, vol. 4595, pp. 24–38. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research. LNCS, vol. 2500. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  11. 11.
    Groce, A., Visser, W.: What went wrong: Explaining counterexamples. In: Ball, T., Rajamani, S.K. (eds.) SPIN 2003. LNCS, vol. 2648, pp. 121–135. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Grumberg, O., Veith, H. (eds.): 25 Years of Model Checking - History, Achievements, Perspectives. LNCS, vol. 5000. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  13. 13.
    Hansen, H., Geldenhuys, J.: Cheap and small counterexamples. In: Cerone, A., Gruner, S. (eds.) SEFM, pp. 53–62. IEEE Computer Society, Los Alamitos (2008)Google Scholar
  14. 14.
    Henzinger, M.R., Telle, J.A.: Faster algorithms for the nonemptiness of Streett automata and for communication protocol pruning. In: Karlsson, R.G., Lingas, A. (eds.) SWAT, pp. 16–27. Springer, Heidelberg (1996)Google Scholar
  15. 15.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-ε. J. Comput. Syst. Sci. 74(3), 335–349 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kupferman, O., Sheinvald-Faragy, S.: Finding shortest witnesses to the nonemptiness of automata on infinite words. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 492–508. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Mitra, R.S.: Strategies for mainstream usage of formal verification. In: Fix, L. (ed.) DAC, pp. 800–805. ACM, New York (2008)CrossRefGoogle Scholar
  18. 18.
    Safra, S.: Complexity of Automata on Infinite Objects. PhD thesis, Weizmann Institute of Science, Rehovot, Israel (March 1989)Google Scholar
  19. 19.
    Schwoon, S., Esparza, J.: A note on on-the-fly verification algorithms. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 174–190. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. In: Proceedings of the VIII Banff Higher order workshop conference on Logics for concurrency: structure versus automata, pp. 238–266. Springer, New York (1996)Google Scholar
  21. 21.
    Wegener, I.: Complexity Theory. In: Exploring the Limits of Efficient Algorithms. Springer, Heidelberg (2004)Google Scholar
  22. 22.
    Woodcock, J., Larsen, P.G., Bicarregui, J., Fitzgerald, J.S.: Formal methods: Practice and experience. ACM Comput. Surv. 41(4) (2009)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Rüdiger Ehlers
    • 1
  1. 1.Reactive Systems GroupSaarland University 

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