Solving first-order constraints in the theory of finite or infinite trees: introduction to the decomposable theories
Résumé
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.