An improved lower bound for the elementary theories of trees

  • Sergei Vorobyov
Session 4A
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1104)


The first-order theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on representing large binary relations by means of short formulas manipulating with high trees, we prove that all the above theories, as well as all their subtheories, are non-elementary in the sense of Kalmar, i.e., cannot be decided within time bounded by a k-story exponential function exp k (n) for any fixed k. Moreover, for some constant d>0 these decision problems require nondeterministic time exceeding exp (⌊dn⌋) infinitely often.


Logic Program Binary Relation Turing Machine Logic Programming Predicate Symbol 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Aït-Kaci, A. Podelski, and G. Smolka. A feature constraint system for logic programming with entailment. Theor. Comput. Sci., 122:263–283, 1994. Preliminary version: 5th Intern. Conf. Fifth Generation Computer Systems, June 1992.CrossRefGoogle Scholar
  2. 2.
    R. Backofen. A complete axiomatization of a theory with feature and arity constraints. J. Logic Programming, 24:37–71, 1995.CrossRefGoogle Scholar
  3. 3.
    R. Backofen and G. Smolka. A complete and recursive feature theory. Theor. Comput. Sci., 146:243–268, 1995. Also: Report DFKI-RR-92-30, 1992.CrossRefGoogle Scholar
  4. 4.
    R. Backofen and R. Treinen. How to win a game with features. In Constraints in Computational Logics'94, volume 845 of Lect. Notes Comput. Sci., pages 320–335. Springer-Verlag, 1994.Google Scholar
  5. 5.
    H. Comon and P. Lescanne. Equational problems and disunification. J. Symb. Computation, 7:371–425, 1989.Google Scholar
  6. 6.
    K. J. Compton and C. W. Henson. A uniform method for proving lower bounds on the computational complexity of logical. theories. Annals Pure Appl. Logic, 48:1–79, 1990.CrossRefGoogle Scholar
  7. 7.
    Yu. L. Ershov, I. A. Lavrov, A. D. Taimanov, and M. A. Taitslin. Elementary theories. Russian Math. Surveys, 20:35–105, 1965.Google Scholar
  8. 8.
    J. Ferrante and C. W. Rackoff. The computational complexity of logical theories, volume 718 of Lect. Notes Math. Springer-Verlag, 1979.Google Scholar
  9. 9.
    M. J. Fisher and M. O. Rabin. Super-exponential complexity of Presburger arithmetic. In SIAM-AMS Proceedings, volume 7, pages 27–41, 1974.Google Scholar
  10. 10.
    W. Hodges. Model Theory, volume 42 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press, 1993.Google Scholar
  11. 11.
    J. Jaffar and M. J. Maher. Constraint logic programming: A survey. J. Logic Programming, 19 & 20:503–581, 1994.CrossRefGoogle Scholar
  12. 12.
    K. Kunen. Answer sets and negation as failure. In J.-L. Lassez, editor, 4th International Conference on Logic Programming, volume 1, pages 219–228. MIT Press, 1987.Google Scholar
  13. 13.
    K. Kunen. Negation in logic programming. J. Logic Programming, 4:289–308, 1987.CrossRefGoogle Scholar
  14. 14.
    M. J. Maher. Complete axiomatizations of the algebras of finite, rational, and infinite trees. In 3rd Annual IEEE Symp. on Logic in Computer Science LICS'88), pages 348–357, 1988.Google Scholar
  15. 15.
    A. I. Malcev. Axiomatizable classes of locally free algebras. In B. F. Wells, editor, The Metamathematics of Algebraic Systems (Collected Papers: 1936–1967), volume 66 of Studies in Logic and the Foundations of Mathematics, chapter 23, pages 262–281. North-Holland Pub. Co., 1971.Google Scholar
  16. 16.
    A. R. Meyer. Weak monadic second-order theory of successor is not elementaryrecursive. In R. Parikh, editor, Logic Colloquium: Symposium on Logic Held at Boston, 1972–1973, volume 453 of Lect. Notes Math., pages 132–154. Springer-Verlag, 1975.Google Scholar
  17. 17.
    P. Odifreddi. Classical recursion theory, volume 125 of Studies in Logic and the Foundations of Mathematics. North-Holland Pub. Co., 1989. Second Edition, 1992.Google Scholar
  18. 18.
    J. I. Seiferas, M. J. Fisher, and A. R. Meyer. Separating nondeterministic time complexity classes. J. ACM, 25(1):146–167, 1978.CrossRefGoogle Scholar
  19. 19.
    G. Smolka. Feature constraint logics for unification grammars. J. Logic Programming, 12:51–87, 1992.CrossRefGoogle Scholar
  20. 20.
    G. Smolka and R. Treinen. Records for logic programming. J. Logic Programming, 18:229–258, 1994. Also: Report DFKI-RR-92-23, 1992.CrossRefGoogle Scholar
  21. 21.
    R. M. Smullyan. Theory of Formal Systems. Princeton University Press, revised edition, 1961.Google Scholar
  22. 22.
    L. J. Stockmeyer. The complexity of decision problems in automata theory and logic. PhD thesis, MIT Lab for Computer Science, 1974. (Also /MIT/LCS Tech Rep 133).Google Scholar
  23. 23.
    L. J. Stockmeyer and A. R. Meyer. Word problems requiring exponential time: preliminary report. In 5th Symp. on Theory of Computing, pages 1–9, 1973.Google Scholar
  24. 24.
    S. Vorobyov. Theory of finite trees revisited: Application of model-theoretic algebra. Technical Report CRIN-94-R-135, Centre de Recherche en Informatique de Nancy, October 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Sergei Vorobyov
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations