Resolution Strategies as Decision Procedures

Abstract

The resolution principle, an automatic inference technique, is studied as a possible decision procedure for certain classes of first-order formulas It is shown that most previous resolution strategies do not decide satlsfiabihty even for "simple" solvable classes Two new resolution procedures are described and are shown to be complete (1 e semidecislon procedures) In the general case and, m addition, to be decision procedures for successively wider classes of first-order formulas These include many previously studied solvable classes The proofs that a complete resolutmn procedure will always halt (without producing the empty clause) when apphed to satisfiable formulas in certain classes provide new, and in some cases more enlightening, demonstrations of the solvablhty of these classes A technique for constructing a model for a formula shown satisfiable in this way is also described KEY WORDS AND PHRASES theorem proving, resolutmn, first-order logic, predicate calculus, decision procedure, solvable case, model, semantic tree CR CATEGORIES 3 60, 5.21, 5 27 IntroductmnMuch of the previous research in automatm theorem proving (here taken to be the study of algorithms for demonstrating the unsatisfiability of first-order formulas) has centered on the resolution method of J.A. Robinson [28]. This technique operates on a set of clauses (each a set of signed atomic formulas) corresponding to a formula of first-order logic. The method is a semideclsion procedure" Given an unsatlsfiable formula, it is guaranteed to generate the empty clause, demonstrating the unsatisfiablhty. By well-known results of Church [5] and Turlng [32], there can be no general dec~swn procedure, which would demonstrate the satisfiabihty or unsatisfiabihty of any first-order formula submitted. There are, however, well-known and easily specified classes of formulas, called solvable classes, for which the question of satisfiabflity or unsatisfiability can be effectively decided; many solvable and unsolvable classes have been extensively investigated.Since Robinson's result, most research in automatm theorem proving has been concerned with the development of strategies for resolutmn which maintain its completeness while attempting to hasten the productmn of the empty clause when given an unsatisfiable formula. Performance of these methods on satisfiable formulas is rarely considered (presumably because they cannot be guaranteed to halt when given such sets of clauses). But complete resolutmn procedures may terminate without generating the empty clause by reaching a point when no new clauses are

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