Abstract
Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable: finite or infinite trees, rational and real numbers, linear dense order,...etc. A decision procedure in the form of five rewriting rules has also been developed. This latter can decide if a first-order formula without free variables is true or not in any decomposable theory. Unfortunately, this decision procedure is not enough when we want to express the solutions of a first-order constraint having free variables. These kind of problems are generally known as first-order constraint satisfaction problems. We present in this paper, not only a decision procedure but a full first-order constraint solver for decomposable theories. Our solver is given in the form of nine rewriting rules which transform any first-order constraint ϕ (which can possibly contain free variables) into an equivalent formula φ which is either the formula true, or the formula false or a simple solved formula having at least one free variable and being equivalent neither to true nor to false. We show the efficiency of our solver by solving complex first-order constraints over finite or infinite trees containing a huge number of imbricated quantifiers and negations and compare the performances with those obtained using the most recent and efficient dedicated solver for finite or infinite trees. This is the first full first-order constraint solver for any decomposable theory.
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References
Apt, K.: Principles of Constraint Programming. Cambridge University Press, Cambridge (2003)
Clark, K.L.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logic and Data Bases. Plenum, New York (1978)
Colmerauer, A., Dao, T.: Expressiveness of full first-order constraints in the algebra of finite or infinite trees. J. Constraints 8(3), 283–302 (2003)
Colmerauer, A.: An introduction to Prolog III. Commun. ACM 33(7), 68–90 (1990)
Dao, T., Djelloul, K.: Solving first-order constraints in the theory of the evaluated trees. In: Recent Advances in Constraints: Revised Selected Paper of the 11th ERCIM International Workshop on Constraint Solving and Contraint Logic Programming. LNAI, vol. 4651, pp. 108–123 (2007)
Djelloul, K.: A full first order constraint solver for decomposable theories. In: Proc of the 9th International Conference of Artificial Intelligence and Symbolic Computation. LNAI, vol. 5144, pp. 93–108 (2008)
Djelloul K., Dao. T., Fruehwirth, T.: Theory of finite or infinite trees revisited. Theory and Practice of Logic Programming (TPLP) 8(4), 431–489 (2008)
Djelloul, K.: Decomposable theories. Theory and Practice of Logic Programming (TPLP) 7(5), 583–632 (2007)
Djelloul, K., Dao, T.: Extension into trees of first-order theories. In: Proc of the 8th International Conference of Artificial Intelligence and Symbolic Computation. LNAI, vol. 4120, pp. 53–67 (2006)
Djelloul, K.: About the combination of trees and rational numbers in a complete first-order theory. In: Proc of the 5th International Symposium on Frontiers of Combining Systems. LNAI, vol. 3717, pp. 106–122 (2005)
Fruehwirth, T., Abdennadher, S.: Essentials of Constraint Programming. Springer, Berlin (2003)
Maher, M.: Complete axiomatizations of the algebras of finite, rational and infinite trees. In: Proc of the 3rd Annual Symposium on Logic in Computer Science, pp. 348–357 (1988)
Oppen, D.: A \(2^{2^{2^n}}\) upper bound on the complexity of Presburger arithmetic. J. Comput. Syst. Sci. 16(3), 323–332 (1978)
Rybina, T., Voronkov, A.: A decision procedure for term algebras with queues. ACM Trans. Comput. Log. 2(2), 155–181 (2001)
Spivey, J.: A categorial approch to the theory of lists. In: Proc of Mathematics of Program Construction, 375th Anniversary of the Groningen University, International Conference. Lecture Notes in Computer Science, vol. 375, pp. 399–408 (1989)
Vorobyov, S.: An improved lower bound for the elementary theories of trees. In: Proc of the 13th International Conference on Automated Deduction. LNAI, vol. 1104, pp. 275–287 (1996)
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Djelloul, K. A full first-order constraint solver for decomposable theories. Ann Math Artif Intell 56, 43–64 (2009). https://doi.org/10.1007/s10472-009-9142-9
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DOI: https://doi.org/10.1007/s10472-009-9142-9