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Unambiguous Büchi Is Weak

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Developments in Language Theory (DLT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9840))

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Abstract

A non-deterministic automaton on infinite trees is unambiguous if it has at most one accepting run on every tree. For a given unambiguous parity automaton \(\mathcal {A} \) of index (i, 2j) we construct an alternating automaton \(\textsc {Transformation}(\mathcal {A})\) which accepts the same language, but is simpler in terms of alternating hierarchy of automata. If \(\mathcal {A} \) is a Büchi automaton (\(i=0, j=1\)), then \(\textsc {Transformation}(\mathcal {A})\) is a weak alternating automaton. In general, \(\textsc {Transformation}(\mathcal {A})\) belongs to the class \(\mathrm {Comp}({i}+1,2{j})\), in particular it is simultaneously of alternating index (i, 2j) and of the dual index \((i+1,2j+1)\). The main theorem of this paper is a correctness proof of the algorithm \(\textsc {Transformation}\). The transformation algorithm is based on a separation algorithm of Arnold and Santocanale (2005) and extends results of Finkel and Simonnet (2009).

The authors were supported by the Polish National Science Centre grant no. 2014-13/B/ST6/03595.

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Notes

  1. 1.

    By Lemma 4 there is at most one such \(\delta \).

References

  1. Arnold, A.: The \(\mu \)-calculus alternation-depth hierarchy is strict on binary trees. ITA 33(4/5), 329–340 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Arnold, A., Santocanale, L.: Ambiguous classes in \(\mu \)-calculi hierarchies. TCS 333(1–2), 265–296 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Bradfield, J.: Simplifying the modal mu-calculus alternation hierarchy. In: Morvan, M., Meinel, C., Krob, D. (eds.) STACS 1998. LNCS, vol. 1373, pp. 39–49. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  4. Carayol, A., Löding, C., Niwiński, D., Walukiewicz, I.: Choice functions and well-orderings over the infinite binary tree. Cent. Eur. J. Math. 8, 662–682 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Colcombet, T.: Forms of determinism for automata (invited talk). In: STACS, pp. 1–23 (2012)

    Google Scholar 

  6. Colcombet, T., Kuperberg, D., Löding, C., Vanden Boom, M.: Deciding the weak definability of Büchi definable tree languages. In: CSL, pp. 215–230 (2013)

    Google Scholar 

  7. Facchini, A., Murlak, F., Skrzypczak, M.: Rabin-Mostowski index problem: a step beyond deterministic automata. In: LICS, pp. 499–508 (2013)

    Google Scholar 

  8. Finkel, O., Simonnet, P.: On recognizable tree languages beyond the Borel hierarchy. Fundam. Informaticae 95(2–3), 287–303 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Hummel, S.: Unambiguous tree languages are topologically harder than deterministic ones. In: GandALF, pp. 247–260 (2012)

    Google Scholar 

  10. Kechris, A.: Classical Descriptive Set Theory. Springer, New York (1995)

    Book  MATH  Google Scholar 

  11. Kupferman, O., Vardi, M.Y.: The weakness of self-complementation. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 455–466. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  12. Murlak, F.: The Wadge hierarchy of deterministic tree languages. Log. Methods Comput. Sci. 4(4), 1–44 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Niwiński, D., Walukiewicz, I.: Ambiguity problem for automata on infinite trees (1996, unpublished)

    Google Scholar 

  14. Niwiński, D., Walukiewicz, I.: Relating hierarchies of word and tree automata. In: Morvan, M., Meinel, C., Krob, D. (eds.) STACS 1998. LNCS, vol. 1373, pp. 320–331. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  15. Niwiński, D., Walukiewicz, I.: A gap property of deterministic tree languages. Theor. Comput. Sci. 1(303), 215–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Niwiński, D., Walukiewicz, I.: Deciding nondeterministic hierarchy of deterministic tree automata. Electr. Notes Theor. Comput. Sci. 123, 195–208 (2005)

    Article  MathSciNet  Google Scholar 

  17. Rabin, M.O.: Weakly definable relations and special automata. In: Proceedings of the Symposium on Mathematical Logic and Foundations of Set Theory, pp. 1–23. North-Holland (1970)

    Google Scholar 

  18. Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 389–455. Springer, Heidelberg (1996)

    Google Scholar 

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Authors and Affiliations

  1. University of Warsaw, Banacha 2, Warsaw, Poland

    Henryk Michalewski & Michał Skrzypczak

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  1. Henryk Michalewski
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  2. Michał Skrzypczak
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Correspondence to Michał Skrzypczak .

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Editors and Affiliations

  1. Université du Québec à Montréal , Montreal, Québec, Canada

    Srečko Brlek

  2. Dept Mathematiques, Univ du Quebec Montreal Dept Mathematiques, Montreal, Québec, Canada

    Christophe Reutenauer

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Michalewski, H., Skrzypczak, M. (2016). Unambiguous Büchi Is Weak. In: Brlek, S., Reutenauer, C. (eds) Developments in Language Theory. DLT 2016. Lecture Notes in Computer Science(), vol 9840. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53132-7_26

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  • DOI: https://doi.org/10.1007/978-3-662-53132-7_26

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