Generalization of Clauses Relative to a Theory

Plotkin's notions of relative θ-subsumption and relative least general generalization of clauses are defined for full clauses, and they are defined in terms of a kind of resolution derivations called C-derivations. Techniques for generalization of clauses relative to a theory, based on the V-operators or saturation in its original form, have primarily been developed for Horn clauses. We show that these techniques are incomplete for full clauses, which is due to the restricted form of resolution derivations considered. We describe a technique for generalization of clauses relative to a theory, which is based on a generalization of the original saturation technique. We prove that our technique properly inverts C-derivations, and that it is complete for full clauses w.r.t. relative θ-subsumption.