Advertisement

Theories of Automatic Structures and Their Complexity

  • Dietrich Kuske
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5725)

Abstract

For automatic structures, several logics have been shown decidable: first-order logic, its extension by the infinity quantifier, by modulo-counting quantifiers, and even by a restricted form of second-order quantification. We review these decidability proofs. As a new result, we determine the data, the expression, and the combined complexity of quantifier-classes for first-order logic. Finally, we also recall that first-order logic becomes elementary decidable for automatic structures of bounded degree.

Keywords

Regular Language Relation Symbol Input Word Bounded Degree Automatic Structure 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bárány, V.: Invariants of automatic presentations and semi-synchronous transductions. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 289–300. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Bárány, V., Kaiser, Ł., Rubin, S.: Cardinality and counting quantifiers on omega-automatic structures. In: STACS 2008, pp. 385–396. IFIB Schloss Dagstuhl (2008)Google Scholar
  3. 3.
    Blumensath, A.: Automatic structures. Technical report, RWTH Aachen (1999)Google Scholar
  4. 4.
    Blumensath, A., Grädel, E.: Automatic Structures. In: LICS 2000, pp. 51–62. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  5. 5.
    Campbell, C.M., Robertson, E.F., Ruškuc, N., Thomas, R.M.: Automatic semigroups. Theoretical Computer Science 250, 365–391 (2001)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Corran, R., Hoffmann, M., Kuske, D., Thomas, R.M.: Singular Artin monoids of finite type are automatic (submitted)Google Scholar
  8. 8.
    Delhommé, Ch., Goranko, V., Knapik, T.: Automatic linear orderings (Manuscript 2003)Google Scholar
  9. 9.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Am. Math. Soc. 98, 21–51 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processin. Groups. Jones and Bartlett Publishers, Boston (1992)Google Scholar
  11. 11.
    Fohry, E., Kuske, D.: On graph products of automatic and biautomatic monoids. Semigroup forum 72, 337–352 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gaifman, H.: On local and nonlocal properties. In: Stern, J. (ed.) Logic Colloquium 1981, pp. 105–135. North-Holland, Amsterdam (1982)Google Scholar
  13. 13.
    Hodgson, B.R.: On direct products of automaton decidable theories. Theoretical Computer Science 19, 331–335 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefGoogle Scholar
  16. 16.
    Khoussainov, B., Nies, A., Rubin, S., Stephan, F.: Automatic structures: richness and limitations. Log. Methods in Comput. Sci. 3(2) (2007)Google Scholar
  17. 17.
    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
  18. 18.
    Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Transactions on Computational Logic 6(4), 675–700 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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
  20. 20.
    Kuske, D., Lohrey, M.: First-order and counting theories of ω-automatic structures. Journal of Symbolic Logic 73, 129–150 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kuske, D., Lohrey, M.: Automatic structures of bounded degree revisited. In: CSL 1999. LNCS. Springer, Heidelberg (to appear, 2009)Google Scholar
  22. 22.
    Kuske, D., Lohrey, M.: Some natural problems in automatic graphs. Journal of Symbolic Logic (accepted, 2009)Google Scholar
  23. 23.
    Lohrey, M.: Automatic structures of bounded degree. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 344–358. Springer, Heidelberg (2003)Google Scholar
  24. 24.
    Oliver, G.P., Thomas, R.M.: Automatic presentations for finitely generated groups. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 693–704. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Rubin, S.: Automata presenting structures: A survey of the finite string case. Bulletin of Symbolic Logic 14, 169–209 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sakarovitch, J.: Easy multiplications. I. The realm of Kleene’s Theorem. Information and Computation 74, 173–197 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Dietrich Kuske
    • 1
  1. 1.Institut für InformatikUniversität LeipzigGermany

Personalised recommendations