Abstract.

The first-order theory of an automatic structure is known to be decidable but there are examples of automatic structures with nonelementary first-order theories. We prove that the first-order theory of an automatic structure of bounded degree (meaning that the corresponding Gaifman-graph has bounded degree) is elementary decidable. More precisely, we prove an upper bound of triply exponential alternating time with a linear number of alternations. We also present an automatic structure of bounded degree such that the corresponding first-order theory has a lower bound of doubly exponential time with a linear number of alternations. We prove similar results also for tree automatic structures.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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 2001), pp. 431–440. IEEE Computer Society Press, Los Alamitos (2001)CrossRefGoogle Scholar
  2. 2.
    Blumensath, A.: Automatic structures. Diploma thesis, RWTH Aachen (1999)Google Scholar
  3. 3.
    Blumensath, A., Grädel, E.: Automatic structures. In: Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science (LICS 2000), pp. 51–62. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  4. 4.
    Campbell, C.M., Robertson, E.F., Ruškuc, N., Thomas, R.M.: Automatic semigroups. Theoretical Computer Science 250(1-2), 365–391 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chandra, K., Kozen, D.C., Stockmeyer, L.J.: Alternation. Journal of the Association for Computing Machinery 28(1), 114–133 (1981)MATHMathSciNetGoogle Scholar
  6. 6.
    Compton, K.J., Henson, C.W.: A uniform method for proving lower bounds on the computational complexity of logical theories. Annals of Pure and Applied Logic 48, 1–79 (1990)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1991)Google Scholar
  8. 8.
    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)MATHGoogle Scholar
  9. 9.
    Ferrante, J., Rackoff, C.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718. Springer, Heidelberg (1979)MATHGoogle Scholar
  10. 10.
    Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. In: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science (LICS 2002), pp. 215–224. IEEE Computer Society Press, Los Alamitos (2002)CrossRefGoogle Scholar
  11. 11.
    Gaifman, H.: On local and nonlocal properties. In: Stern, J. (ed.) Logic Colloquium 1981, pp. 105–135. North Holland, Amsterdam (1982)Google Scholar
  12. 12.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)MATHCrossRefGoogle Scholar
  13. 13.
    Hodgson, B.R.: On direct products of automaton decidable theories. Theoretical Computer Science 19, 331–335 (1982)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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
  15. 15.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)Google Scholar
  16. 16.
    Khoussainov, B., Rubin, S.: Graphs with automatic presentations over a unary alphabet. Journal of Automata, Languages and Combinatorics 6(4), 467–480 (2001)MATHMathSciNetGoogle Scholar
  17. 17.
    Khoussainov, B., Rubin, S., Stephan, F.: Automatic partial orders. In: Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science, LICS 2003 (2003) (to appear)Google Scholar
  18. 18.
    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
  19. 19.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  20. 20.
    Silva, P.V., Steinberg, B.: A geometric characterization of automatic monoids. Technical Report CMUP 2000-03, University of Porto (2001)Google Scholar
  21. 21.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time (preliminary report). In: Proceedings of the 5th Annual ACM Symposium on Theory of Computing (STOCS 1973), pp. 1–9. ACM Press, New York (1973)CrossRefGoogle 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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Markus Lohrey
    • 1
  1. 1.Institut für Formale Methoden der InformatikUniversität StuttgartStuttgartGermany

Personalised recommendations