Theory of Finite or Infinite Trees Revisited

Khalil Djelloul (1) , Thi-Bich-Hanh Dao (1) , Thom Fruehwirth (2)

1 LIFO - Laboratoire d'Informatique Fondamentale d'Orléans
2 Universität Ulm - Ulm University [Ulm, Allemagne]

We present in this paper a first-order axiomatization of an extended theory T of finite or infinite trees, built on a signature containing an infinite set of function symbols and a relation finite(t) which enables to distinguish between finite or infinite trees. We show that T has at least one model and prove its completeness by giving not only a decision procedure, but a full first-order constraint solver which gives clear and explicit solutions for any first-order constraint satisfaction problem in T. The solver is given in the form of 16 rewriting rules which transform any first-order constraint ' into an equivalent disjunction D of simple formulas such that D is either the formula true or the formula false or a formula having at least one free variable, being equivalent neither to true nor to false and where the solutions of the free variables are expressed in a clear and explicit way. The correctness of our rules implies the completeness of T. We also describe an implementation of our algorithm in CHR (Constraint Handling Rules) and compare the performance with an implementation in C++ and that of a recent decision procedure for decomposable theories.

Domains: Logic in Computer Science [cs.LO]

Khalil Djelloul, Thi-Bich-Hanh Dao, Thom Fruehwirth. Theory of Finite or Infinite Trees Revisited. Theory and Practice of Logic Programming, 2008, pp.8(4):1-60. ⟨hal-00202312⟩