# An improved lower bound for the elementary theories of trees

## Abstract

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.

## Keywords

Logic Program Binary Relation Turing Machine Logic Programming Predicate Symbol## Preview

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## References

- 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.R. Backofen. A complete axiomatization of a theory with feature and arity constraints.
*J. Logic Programming*, 24:37–71, 1995.CrossRefGoogle Scholar - 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.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.H. Comon and P. Lescanne. Equational problems and disunification.
*J. Symb. Computation*, 7:371–425, 1989.Google Scholar - 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.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.J. Ferrante and C. W. Rackoff.
*The computational complexity of logical theories*, volume 718 of*Lect. Notes Math.*Springer-Verlag, 1979.Google Scholar - 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.W. Hodges.
*Model Theory*, volume 42 of*Encyclopedia of Mathematics and its Applications*. Cambridge Univ. Press, 1993.Google Scholar - 11.J. Jaffar and M. J. Maher. Constraint logic programming: A survey.
*J. Logic Programming*, 19 & 20:503–581, 1994.CrossRefGoogle Scholar - 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.K. Kunen. Negation in logic programming.
*J. Logic Programming*, 4:289–308, 1987.CrossRefGoogle Scholar - 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.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.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.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.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.G. Smolka. Feature constraint logics for unification grammars.
*J. Logic Programming*, 12:51–87, 1992.CrossRefGoogle Scholar - 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.R. M. Smullyan.
*Theory of Formal Systems*. Princeton University Press, revised edition, 1961.Google Scholar - 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.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.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