On Finitely Ambiguous Büchi Automata

  • Christof Löding
  • Anton Pirogov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11088)


Unambiguous Büchi automata, i.e. Büchi automata allowing only one accepting run per word, are a useful restriction of Büchi automata that is well-suited for probabilistic model-checking. In this paper we propose a more permissive variant, namely finitely ambiguous Büchi automata, a generalisation where each word has at most k accepting runs, for some fixed k. We adapt existing notions and results concerning finite and bounded ambiguity of finite automata to the setting of \(\omega \)-languages and present a translation from arbitrary nondeterministic Büchi automata with n states to finitely ambiguous automata with at most \(3^n\) states and at most n accepting runs per word.


Büchi automata Infinite words Ambiguity 


  1. 1.
    Allauzen, C., Mohri, M., Rastogi, A.: General algorithms for testing the ambiguity of finite automata. In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257, pp. 108–120. Springer, Heidelberg (2008). Scholar
  2. 2.
    Arnold, A.: Rational \(\omega \)-languages are non-ambiguous. Theor. Comput. Sci. 26(1–2), 221–223 (1983)CrossRefGoogle Scholar
  3. 3.
    Baier, C., Katoen, J.: Principles of Model Checking. MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  4. 4.
    Baier, C., Kiefer, S., Klein, J., Klüppelholz, S., Müller, D., Worrell, J.: Markov chains and unambiguous Büchi automata. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9779, pp. 23–42. Springer, Cham (2016). Scholar
  5. 5.
    Bousquet, N., Löding, C.: Equivalence and inclusion problem for strongly unambiguous Büchi automata. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 118–129. Springer, Heidelberg (2010). Scholar
  6. 6.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Studies in Logic and the Foundations of Mathematics, vol. 44, pp. 1–11. Elsevier (1966)Google Scholar
  7. 7.
    Chan, T.H., Ibarra, O.H.: On the finite-valuedness problem for sequential machines. Theor. Comput. Sci. 23(1), 95–101 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Couvreur, J.-M., Saheb, N., Sutre, G.: An optimal automata approach to LTL model checking of probabilistic systems. In: Vardi, M.Y., Voronkov, A. (eds.) LPAR 2003. LNCS (LNAI), vol. 2850, pp. 361–375. Springer, Heidelberg (2003). Scholar
  10. 10.
    Isaak, D., Löding, C.: Efficient inclusion testing for simple classes of unambiguous \(\omega \)-automata. Inf. Process. Lett. 112(14–15), 578–582 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kähler, D., Wilke, T.: Complementation, disambiguation, and determinization of Büchi automata unified. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 724–735. Springer, Heidelberg (2008). Scholar
  12. 12.
    Karmarkar, H., Joglekar, M., Chakraborty, S.: Improved upper and lower bounds for Büchi disambiguation. In: Van Hung, D., Ogawa, M. (eds.) ATVA 2013. LNCS, vol. 8172, pp. 40–54. Springer, Cham (2013). Scholar
  13. 13.
    Leung, H.: Separating exponentially ambiguous finite automata from polynomially ambiguous finite automata. SIAM J. Comput. 27(4), 1073–1082 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Safra, S.: On the complexity of omega-automata. In: Foundations of Computer Science 29th Annual Symposium on 1988, pp. 319–327. IEEE (1988)Google Scholar
  15. 15.
    Stearns, R.E., Hunt III, H.B.: On the equivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata. SIAM J. Comput. 14(3), 598–611 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Thomas, W.: Languages, Automata, and Logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 389–455. Springer, Heidelberg (1997). Scholar
  17. 17.
    Weber, A., Seidl, H.: On the degree of ambiguity of finite automata. Theor. Comput. Sci. 88(2), 325–349 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.RWTH Aachen UniversityAachenGermany

Personalised recommendations