Abstract
We are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relation symbol and function symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly infinite number, are labelled by elements of F. The operation linked to each element f of F is the mapping (a 1,..., a n ) ↦ b, where b is the tree whose initial node is labelled f and whose sequence of daughters is a 1,..., a n .
We first consider tree constraints involving long alternated sequences of quantifiers ∃∀∃∀.... We show how to express winning positions of two-person games with such constraints and apply our results to two examples.
We then construct a family of strongly expressive tree constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number α(k), obtained by top down evaluating a power tower of 2's, of height k. By a tree constraint of size proportional to k, it is then possible to define a tree having exactly α(k) nodes or to express the multiplication table computed by a Prolog machine executing up to α(k) instructions.
By replacing the Prolog machine with a Turing machine we show the quasi-universality of tree constraints, that is to say, the ability to concisely describe trees which the most powerful machine will never have time to compute. We also rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a tree constraint without free variables is true, cannot be bounded above by a function obtained from finite composition of simple functions including exponentiation.
Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generalities, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we notice that it is possible to solve a constraint of 5000 symbols involving 160 alternating quantifiers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benhamou, F., Bouvier, P., Colmerauer, A., Garetta, H., Giletta, B., Massat, J. L., Narboni, G. A., N'Dong, S., Pasero, R., Pique, J. F., Touraώvane, Van Caneghem, M., & VÉtillard, E. (1996). Le manuel de Prolog IV. PrologIA: Marseille.
Clark, K. L. (1978). Negation as failure. In Gallaire, H., & Minker, J., eds., Logic and Databases, pages 293-322. New York: Plenum Press.
Colmerauer, A. (1982). Prolog and infinite trees, In Clark, K. L., & Tarnlund, S. A., eds., Logic Programming, pages 231-251. Academic Press: New York.
Colmerauer, A., Kanoui, Henry, & Van Caneghem, Michel. Prolog, theoretical principles and current trends, In Technology and Science of Informatics, North Oxford Academic, vol. 2, no 4, August 1983. English version of the journal TSI, AFCET-Bordas, where the paper appears under the title: Prolog, bases théoriques et développements actuels.
Colmerauer, A. (1984). Equations and inequations on finite and infinite trees. In Proceeding of the International Conference on Fifth Generation Computer Systems (FCGS-84), pages 85-99. Tokyo: ICOT.
Colmerauer, A. (1990). An introduction to prolog III. Communications of the ACM, 33(7): 68-90.
Coupet-Grimal, S., & Ridoux, O. On the use of advanced logic programming features in computational linguistics. The Journal of Logic Programming, 24(1&2): 121-159.
Courcelle, B. (1983). Fundamental Properties Of Infinite Trees. Theoretical Computer Science, 25(2): 95-169 (March).
Courcelle, B. (1986). Equivalences and transformations of regular systems-applications to program schemes and grammars. Theoretical Computer Science, 42: 1-122.
Dao, Thi-Bich-Hanh. (2000). Résolution de contraintes du premier ordre dans la théorie des arbres finis ou infinis. In Programmation en logique avec contraintes, JFPLC'2000, pages 225-240. Hermés Sciences et Publication.
Dao, Thi-Bich-Hanh. (2000). Résolution de contraintes du premier ordre dans la théorie des arbres finis ou infinis. PhD thesis, Université de la Méditerranée (December).
Huet, G. (1976). Résolution d'équations dans les langages d'ordre 1,2, ⋯,w, Thése d'Etat, Université Paris 7.
Maher, M. J. (1988). Complete Axiomatization of the Algebra of Finite, Rational and Infinite Trees. Technical report, IBM-T.J. Watson Research Center.
Mielniczuk, P. Basic Theory of Feature Trees, submitted to Journal of Symbolic Computation, available at http://www.tcs.uni.wroc.pl/fmielni.
Sipser, M. (1997). Introduction to the Theory of Computation, PWS Publishing Company.
Vorobyov, S. (1996). An Improved Lower Bound for the Elementary Theories of Trees. In Proceeding of the 13th International Conference on Automated Deduction (CADE'96), Springer Lecture Notes in Artificial Intelligence, Vol. 1104, pages 275-287. New Brunswick, NJ (July/August).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Colmerauer, A., Dao, TBH. Expressiveness of Full First-Order Constraints in the Algebra of Finite or Infinite Trees. Constraints 8, 283–302 (2003). https://doi.org/10.1023/A:1025675127871
Issue Date:
DOI: https://doi.org/10.1023/A:1025675127871