Skip to main content
SpringerLink
Log in

Automatic Structures of Bounded Degree

  • Conference paper
Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2003)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2850))

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)

    Chapter  Google Scholar 

  2. Blumensath, A.: Automatic structures. Diploma thesis, RWTH Aachen (1999)

    Google Scholar 

  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. Campbell, C.M., Robertson, E.F., Ruškuc, N., Thomas, R.M.: Automatic semigroups. Theoretical Computer Science 250(1-2), 365–391 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chandra, K., Kozen, D.C., Stockmeyer, L.J.: Alternation. Journal of the Association for Computing Machinery 28(1), 114–133 (1981)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1991)

    Google Scholar 

  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)

    MATH  Google Scholar 

  9. Ferrante, J., Rackoff, C.: The Computational Complexity of Logical Theories. Lecture Notes in Mathematics, vol. 718. Springer, Heidelberg (1979)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  13. Hodgson, B.R.: On direct products of automaton decidable theories. Theoretical Computer Science 19, 331–335 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Khoussainov, B., Rubin, S.: Graphs with automatic presentations over a unary alphabet. Journal of Automata, Languages and Combinatorics 6(4), 467–480 (2001)

    MATH  MathSciNet  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  19. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  20. Silva, P.V., Steinberg, B.: A geometric characterization of automatic monoids. Technical Report CMUP 2000-03, University of Porto (2001)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Institut für Formale Methoden der Informatik, Universität Stuttgart, Universitätsstr. 38, 70569, Stuttgart, Germany

    Markus Lohrey

Authors
  1. Markus Lohrey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Rice University, 77251-1892, Houston, TX, U.S.A.

    Moshe Y. Vardi

  2. University of Manchester, UK

    Andrei Voronkov

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lohrey, M. (2003). Automatic Structures of Bounded Degree. In: Vardi, M.Y., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2003. Lecture Notes in Computer Science(), vol 2850. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39813-4_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-39813-4_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20101-4

  • Online ISBN: 978-3-540-39813-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation