Non-Safe SLDNF Resolution and Least Fixpoint Semantics for General Logic Programs

Abstract

The SLDNF resolution (SLD resolution with negation as failure) is often restricted to yield a safe rule, that is, negation as failure rule is adopted only the case the selected negative literal in each goal should be in ground. To investigate the application of SLDNF resolution to the case of selecting non-ground negative literals, we deal with the success and failure sets by non-safe SLDNF resolution as well as its mathematical property. Since Shepherdson's proposal is thought of as most general [Shepherdson 85], we firstly formulate Shepherdson's SLDNFS refutation as the relation on the set of goals and the set of answers, and his finite failure deduction as the relation on the set of goals. Then, as the purpose of the paper, we characterize the denotation of the success and failure sets by the SLDNF resolution with a fixpoint semantics, which is generalized to be concerned with atom sets the variables occur in. For the purpose, we show the pair of success and failure sets is included in the least fixpoint of a given general logic program. The inclusion is regarded as a generalization of the equivalance between the success set and the least fixpoint semantics for a set of definite clauses. Also the inclusion should be shown in non-ground atom set version, extending the abstract interpretation of the success and failure sets with respect to the least fixpoint in the ground version.