Chapter 5 - Reasoning with Horn Clauses

Abstract

This chapter explores the idea of limitation that are applicable to a certain interesting subset of first-order logic, where the Resolution procedure becomes much more manageable. In a Resolution-based system, clauses end up being used for two different purposes. Full first-order logic is concerned with disjunction and incomplete knowledge in a more general form that are putting aside for the purposes of this chapter. A Resolution derivation over Horn clauses, observes that two negative clauses can never be resolved together, because all of their literals are of the same polarity. If one resolves a negative and a positive clause together, it is guaranteed to produce a negative clause: The two clauses must be resolved with respect to the one positive literal in the positive clause, and so it does not appear in the resolvent. Similarly, if one resolves two positive clauses together, it is guaranteed to produce a positive clause: The two clauses must be resolved with respect to one (and only one) of the positive literals, so the other positive literal appears in the resolvent. In other words, Resolution over Horn clauses must always involve a positive clause, and if the second clause is negative, the resolvent is negative; if the second clause is positive, the resolvent is positive.