Compulsory Reduction in Linear Derivation Systems

This letter clarifies the issue of compulsory reduction in linear derivation systems, and will be a helpful warning to new researchers in this area. Well known linear derivation systems include the ME procedure [3], SL-resolution [2], OL deduction [1], and the GC procedure [6]. Each of these systems incorporates a reduction operation to implement ancestor resolution and, in the GC procedure, a retrospective form of factoring. Reduction can be made compulsory if the literals involved have identical atoms, but generalising compulsory reduction to apply when the atoms are unifiable (not necessarily identical) is an incomplete strategy. This point deserves clarification, as it has been inadequately presented in the literature. There are three cases of particular interest : (i) The discussion that follows the formal definition of SL-resolution [2, pg 238] states that "...if a literal can be removed by reduction, then this is done before any extension operations are performed". The obvious misinterpretation of this statement is that if both extension and reduction are possible, then reduction is the compulsory choice. SL-resolution's reduction is in fact only compulsory if the atoms of the literals in question are, or become, identical. (ii) As presented in [1], OL deduction specifies that "If C i is a reducible ordered clause, then C i+1 is the reduced ordered clause of C i ". This restriction makes OL deduction incomplete, and the flaw in the associated completeness theorem (Theorem 7.2) was noted in [4]. This error persists in the 1987 reprint of [1]. (iii) [6] defines the GC procedure at the ground level, in which "C * i+1 must be obtained from C * i by reduction, if reduction is possible", and goes on to say that the "ground procedure lifts in a straightforward manner to the first-order level". [5, pg 6-51] notes that "on the general level, reduction cannot be compulsory", but this is not mentioned in [6]. To conclude, in the above mentioned and other linear derivation systems, compulsory reduction is in general not a complete strategy. A counterexample for compulsory reduction against A-literals is S = {r(c), p v ~r(c), ~p v q(X) v ~r(X), r(b), ~q(b)} with r(c) as the top clause, and for compulsory reduction against C-literals in the GC procedure, S = {~p(X) v ~q(X) v ~r, r v ~q(b), q(b), q(a), p(a)} with ~p(X) v ~q(X) v ~r as the top clause. The selection function that chooses the …