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First-Order and Counting Theories of ω-Automatic Structures

  • Dietrich Kuske
  • Markus Lohrey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3921)

Abstract

The logic \({\mathcal L}({\mathcal Q}_u)\) extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying ... belongs to the set C”). This logic is investigated for structures with an injective ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [4]. It is shown that, as in the case of automatic structures [13], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are \(\varkappa\) many elements satisfying ...”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of \({\mathcal L}({\mathcal Q}_u)\) that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.

Keywords

Decidable Theory Turing Machine Relational Symbol Annual IEEE Symposium Bounded Degree 
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.

References

  1. 1.
    Benedikt, M., Libkin, L., Schwentick, T., Segoufin, L.: A model-theoretic approach to regular string relations. In: Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (LICS 2000), pp. 431–440. IEEE Computer Society Press, Los Alamitos (2001)CrossRefGoogle Scholar
  2. 2.
    Bès, A.: Undecidable extensions of Büchi arithmetic and Cobham-Semenov theorem. Journal of Symbolic Logic 62(4), 1280–1296 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blumensath, A.: Automatic structures. Diploma thesis, RWTH Aachen (1999)Google Scholar
  4. 4.
    Blumensath, A., Grädel, E.: Finite presentations of infinite structures: Automata and interpretations. Theory of Computing Systems 37(6), 641–674 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Büchi, R.: Weak second-order arithmetic and finite automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 6, 66–92 (1960)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word processing in groups, Jones and Bartlett, Boston (1992)Google Scholar
  7. 7.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)CrossRefMATHGoogle Scholar
  8. 8.
    Hodgson, B.R.: On direct products of automaton decidable theories. Theoretical Computer Science 19, 331–335 (1982)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ishihara, H., Khoussainov, B., Rubin, S.: Some results on automatic structures. In: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science (LICS 2002), pp. 235–244. IEEE Computer Society Press, Los Alamitos (2002)CrossRefGoogle Scholar
  10. 10.
    Keisler, H.J., Lotfallah, W.B.: A local normal form theorem for infinitary logic with unary quantifiers. Mathematical Logic Quarterly 51(2), 137–144 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  12. 12.
    Khoussainov, B., Rubin, S., Stephan, F.: Automatic partial orders. In: Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science (LICS 2003), pp. 168–177. IEEE Computer Society Press, Los Alamitos (2003)CrossRefGoogle Scholar
  13. 13.
    Khoussainov, B., Rubin, S., Stephan, F.: Definability and regularity in automatic structures. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 440–451. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Kuske, D.: Is Cantor’s theorem automatic. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 332–345. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Kuske, D., Lohrey, M.: First-order and counting theories of ω-automatic structures. Technical Report 2005-07, Universität Stuttgart (2005), available at: ftp://ftp.informatik.uni-stuttgart.de/pub/library/ncstrl.ustuttgart_fi/TR-2005-07
  16. 16.
    Lohrey, M.: Automatic structures of bounded degree. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 346–360. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Perrin, D., Pin, J.-E.: Infinite Words. In: Pure and Applied Mathematics, Elsevier, Amsterdam (2004)Google Scholar
  18. 18.
    Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch.4, pp. 133–191. Elsevier Science Publishers B. V, Amsterdam (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Dietrich Kuske
    • 1
  • Markus Lohrey
    • 2
  1. 1.Institut für InformatikUniversität LeipzigGermany
  2. 2.Universität Stuttgart, FMIGermany

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