Journal of Logic Programming
Equational problems anddisunification
Journal of Symbolic Computation
Communications of the ACM
A Machine-Oriented Logic Based on the Resolution Principle
Journal of the ACM (JACM)
An Efficient Unification Algorithm
ACM Transactions on Programming Languages and Systems (TOPLAS)
Incremental Algorithms for Constraint Solving and Entailment over Rational Trees
Proceedings of the 13th Conference on Foundations of Software Technology and Theoretical Computer Science
Solving Disequations in Equational Theories
Proceedings of the 9th International Conference on Automated Deduction
An Improved Lower Bound for the Elementary Theories of Trees
CADE-13 Proceedings of the 13th International Conference on Automated Deduction: Automated Deduction
About the combination of trees and rational numbers in a complete first-order theory
FroCoS'05 Proceedings of the 5th international conference on Frontiers of Combining Systems
Theory and Practice of Logic Programming
Theory of finite or infinite trees revisited
Theory and Practice of Logic Programming
Solving first-order constraints in the theory of the evaluated trees
CSCLP'06 Proceedings of the constraint solving and contraint logic programming 11th annual ERCIM international conference on Recent advances in constraints
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We present in this paper an algorithm in the theory T of possibly infinite trees for solving general constraints represented by full first-order formulas, with equality as the only relation and functional symbols taken from an infinite set F. The algorithm consists of a set of 11 rewriting rules. It transforms a first-order formula p in a conjunction q of solved formulas, equivalent in T, such that: (1) the conjunction q is the formula true if p is always true in T, and the formula ¬true if p is always false in T. Moreeover, if p or its negation has a finite base of solutions in a model of T, then these solutions or non-solutions have to be explicit in q. (2) each solved formula of q has not new free variables and can be transformed immediately in a Boolean combination of basic formulas whose length does not exceed twice the length of the solved formula. The basic formulas are particular cases of existentially quantified conjunctions of equations. The correctness of the algorithm gives another proof of the completeness of T demonstrated by Michael Maher. We test our algorithm on benchmarks realized by an implementation, solving formulas on two-partners games in T with more than 160 nested alternated quantifiers.Finally, we show then that we can generalize this algorithm by introducing a new class of theories that we call decomposable. We show that T is decomposable and give a general algorithm for solving first-order constraint in any decomposable theory. The correctness of our algorithm shows the completeness of the decomposable theories.