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Complementation of Finitely Ambiguous Büchi Automata

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Developments in Language Theory (DLT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11088))

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Abstract

A nondeterministic automaton is finitely ambiguous if for each input there is at most finitely many accepting runs. We prove that the complement of the \(\omega \)-language accepted by a finitely ambiguous Büchi automaton with n states is accepted by an unambiguous Büchi automaton with \(2\times 5^n\) states.

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Notes

  1. 1.

    Unfortunately, an automaton on words is called finitely ambiguous if it is k-ambiguous for some k. Maybe a more appropriate name for such automata is “bounded ambiguous”.

  2. 2.

    A state is useful if it is on an accepting run.

References

  1. Arnold, A.: Rational \(\omega \)-languages are non-ambiguous. Theor. Comput. Sci. 26, 221–223 (1983)

    Article  Google Scholar 

  2. Baier, C., Kiefer, S., Klein, J., Klüppelholz, S., Müller, D., Worrell, J.: Markov chains and unambiguous Büchi automata. In: CAV, pp. 23–42 (2016)

    Google Scholar 

  3. Büchi, J.R.: On a decision method in restricted second order arithmetic. In: CLMPS 1960, pp. 1–11. Stanford University Press (1962)

    Google Scholar 

  4. Colcombet, T.: Unambiguity in automata theory. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 3–18. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19225-3_1

    Chapter  Google Scholar 

  5. Friedgut, E., Kupferman, O., Vardi, M.Y.: Büchi complementation made tighter. Int. J. Found. Comput. Sci. 17(4), 851–868 (2006)

    Article  Google Scholar 

  6. Han, Y., Salomaa, A., Salomaa, K.: Ambiguity, nondeterminism and state complexity of finite automata. Acta Cybern. 23(1), 141–157 (2017)

    Article  MathSciNet  Google Scholar 

  7. Isaak, D., Loding, C.: Efficient inclusion testing for simple classes of unambiguous \(\omega \)-automata. IPL 112(14–15), 578–582 (2012)

    Article  MathSciNet  Google Scholar 

  8. 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). https://doi.org/10.1007/978-3-540-70575-8_59

    Chapter  MATH  Google Scholar 

  9. Leiss, E.: Succinct representation of regular languages by Boolean automata. Theor. Comput. Sci. 13, 323–330 (1981)

    Article  MathSciNet  Google Scholar 

  10. McNaughton, R.: Testing and generating infinite sequences by a infnite automaton. Inf. Control 9(5), 521–530 (1966)

    Article  Google Scholar 

  11. Miyano, S., Hayashi, T.: Alternating Infite automata on \(\omega \)-words. Theor. Comput. Sci. 32(3), 321–330 (1984)

    Article  Google Scholar 

  12. Leung, H.: Descriptional complexity of NFA of different ambiguity. Int. J. Found. Comput. Sci. 16(5), 975–984 (2005)

    Article  MathSciNet  Google Scholar 

  13. Jirásek, J., Jirásková, G., Šebej, J.: Operations on unambiguous finite automata. In: Brlek, S., Reutenauer, C. (eds.) DLT 2016. LNCS, vol. 9840, pp. 243–255. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53132-7_20

    Chapter  Google Scholar 

  14. Safra, S.: On the complexity of omega-automata. In: FOCS 1988, pp. 319–327. IEEE Computer Society (1988)

    Google Scholar 

  15. Schewe, S.: Büchi complementation made tight. In: TACS 2009, pp. 661–672 (2009)

    Google Scholar 

  16. Sistla, A.P., Vardi, M.Y., Wolper, P.: The complementation problem for Büchi automata with applications to temporal logic. TCS 49(3), 217–239 (1987)

    Article  Google Scholar 

  17. Stearns, R.E., Hunt, H.B.: On the equivalence and containment problems for un-ambiguous regular expressions, regular grammars and finite automata. SIAM J. Comput. 14(3), 598–611 (1985)

    Article  MathSciNet  Google Scholar 

  18. Thomas, W.: Automata on infinite objects. In: Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B, pp. 133–192 (1990)

    Google Scholar 

  19. Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: FOCS 1985, pp. 327–338 (1985)

    Google Scholar 

  20. Weber, A., Seidl, H.: On the degree of ambiguity of finite automata. Theor. Comput. Sci. 88(2), 325–349 (1991)

    Article  MathSciNet  Google Scholar 

  21. Yan, Q.: Lower bounds for complementation of omega-automata via the full automata technique. J. Log. Methods Comput. Sci. 4(1:5) (2008)

    Google Scholar 

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Acknowledgments

I would like to thank anonumous reviewers for their useful and insightful suggestions.

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Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Alexander Rabinovich

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  1. Alexander Rabinovich
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Correspondence to Alexander Rabinovich .

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Editors and Affiliations

  1. National Archives of Japan, Tokyo, Japan

    Mizuho Hoshi

  2. The University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

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Rabinovich, A. (2018). Complementation of Finitely Ambiguous Büchi Automata. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_44

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  • DOI: https://doi.org/10.1007/978-3-319-98654-8_44

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