Feasible Automata for Two-Variable Logic with Successor on Data Words

  • Ahmet Kara
  • Thomas Schwentick
  • Tony Tan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7183)

Abstract

We introduce an automata model for data words, that is words that carry at each position a symbol from a finite alphabet and a value from an unbounded data domain. The model is (semantically) a restriction of data automata, introduced by Bojanczyk, et. al. in 2006, therefore it is called weak data automata. It is strictly less expressive than data automata and the expressive power is incomparable with register automata. The expressive power of weak data automata corresponds exactly to existential monadic second order logic with successor + 1 and data value equality ~, EMSO2( + 1,~). It follows from previous work, David, et. al. in 2010, that the nonemptiness problem for weak data automata can be decided in 2-NEXPTIME. Furthermore, we study weak Büchi automata on data ω-strings. They can be characterized by the extension of EMSO2( + 1,~) with existential quantifiers for infinite sets. Finally, the same complexity bound for its nonemptiness problem is established by a nondeterministic polynomial time reduction to the nonemptiness problem of weak data automata.

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References

  1. 1.
    Björklund, H., Schwentick, T.: On notions of regularity for data languages. Theor. Comput. Sci. 411(4-5), 702–715 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boasson, L.: Some applications of CFL’s over infinte alphabets. Theoretical Computer Science, 146–151 (1981)Google Scholar
  3. 3.
    Bojanczyk, M., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on data trees and XML reasoning. J. ACM 56(3) (2009)Google Scholar
  4. 4.
    Bojanczyk, M., Muscholl, A., Schwentick, T., Segoufin, L., David, C.: Two-variable logic on words with data. In: LICS, pp. 7–16 (2006)Google Scholar
  5. 5.
    Bollig, B.: An automaton over data words that captures EMSO logic. CoRR abs/1101.4475 (2011)Google Scholar
  6. 6.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundl. Math. 6, 66–92 (1960)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cheng, E.Y.C., Kaminski, M.: Context-free languages over infinite alphabets. Acta Inf. 35(3), 245–267 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Colcombet, T., Ley, C., Puppis, G.: On the Use of Guards for Logics with Data. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 243–255. Springer, Heidelberg (2011)Google Scholar
  9. 9.
    David, C., Libkin, L., Tan, T.: On the Satisfiability of Two-Variable Logic over Data Words. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 248–262. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Demri, S., D’Souza, D., Gascon, R.: A Decidable Temporal Logic of Repeating Values. In: Artemov, S., Nerode, A. (eds.) LFCS 2007. LNCS, vol. 4514, pp. 180–194. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Demri, S., Lazic, R.: LTL with the freeze quantifier and register automata. ACM Trans. Comput. Log. 10(3) (2009)Google Scholar
  12. 12.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Transactions of The American Mathematical Society 98, 21 (1961)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gischer, J.L.: Shuffle languages, Petri nets, and context-sensitive grammars. Commun. ACM 24(9), 597–605 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Grädel, E., Otto, M.: On logics with two variables. Theor. Comput. Sci. 224(1-2), 73–113 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kaminski, M., Francez, N.: Finite-memory automata. Theor. Comput. Sci. 134(2), 329–363 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kaminski, M., Tan, T.: Regular expressions for languages over infinite alphabets. Fundam. Inform. 69(3), 301–318 (2006)MathSciNetMATHGoogle Scholar
  17. 17.
    Kara, A., Schwentick, T., Tan, T.: Feasible automata for two-variable logic with successor on data words, arXiv:1110.1221v1Google Scholar
  18. 18.
    Lazic, R.: Safety alternating automata on data words. ACM Trans. Comput. Log. 12(2), 10 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Neven, F., Schwentick, T., Vianu, V.: Finite state machines for strings over infinite alphabets. ACM Trans. Comput. Log. 5(3), 403–435 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Niewerth, M., Schwentick, T.: Two-variable logic and key constraints on data words. In: ICDT, pp. 138–149 (2011)Google Scholar
  21. 21.
    Otto, F.: Classes of regular and context-free languages over countably infinite alphabets. Discrete Applied Mathematics 12(1), 41–56 (1985)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. III, pp. 389–455. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  23. 23.
    Trakhtenbrot, B.: Finite automata and logic of monadic predicates. Doklady Akademii Nauk SSSR 140, 326–329 (1961)Google Scholar
  24. 24.
    Wu, Z.: A decidable extension of data automata. In: GandALF, pp. 116–130 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ahmet Kara
    • 1
  • Thomas Schwentick
    • 1
  • Tony Tan
    • 2
  1. 1.Technical University of DortmundGermany
  2. 2.University of EdinburghUK

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