Completeness and Redundancy in Constrained Clause Logic

Abstract

. In [6], a resolution-based inference system on c-clauses (i.e. constrained clauses) was introduced, incorporating powerful deletion rules for redundancy elimination. This inference system was extended to resolution refinements in subsequent papers of Caferra et al. (e.g. [4] and [5]). The completeness proofs given for the purely refutational calculi (i.e.: the inference systems without deletion rules) are basically translations of the corresponding results from standard clause logic to constrained clause logic (= c-clause logic, for short). This work focuses on the deletion rules of the calculi of Caferra et al. and, in particular, on the c-dissubsumption rule, which is considerably more powerful than the usual subsumption concept in standard clause logic. We will show that the conventional method for proving the completeness of (standard clause) resolution refinements with subsumption fails when the powerful deletion rules of Caferra et al. are considered. Therefore, in order to prove the completeness of the c-clause calculi, a different strategy is required. To this end, we shall extend the well-known concept of semantic trees from standard clause logic to c-clause logic. In general, purely non-deterministic application of the inference rules is not sufficient to ensure refutational completeness. It is intuitively clear, that some sort of fairness must be required. The completeness proof via semantic trees gives us a hint for defining precisely what it means for a rule application strategy to be fair. Finally other methods for proving completeness and defining redundancy criteria are contrasted with completeness via semantic trees and c-dissubsumption. In particular, it is shown that the redundancy criteria within the ordering-based approaches of Bachmair/Ganzinger (cf. [2]) and Nieuwenhuis/Rubio (cf. [11]) are incomparable with c-dissubsumption.