Elementary Sets for Logic Programs

TL;DR

This paper introduces the concept of elementary sets in logic programming, simplifying and extending the notion of elementary loops, and provides new theoretical insights and characterizations for both nondisjunctive and disjunctive programs.

Contribution

It generalizes the notion of elementary loops to elementary sets, making it applicable to disjunctive programs and offering simpler characterizations and complexity results.

Findings

Elementary sets are almost equivalent to elementary loops for nondisjunctive programs.

Maximal unfounded elementary sets correspond to minimal nonempty unfounded sets.

Deciding elementary sets is coNP-complete for disjunctive programs.

Abstract

By introducing the concepts of a loop and a loop formula, Lin and Zhao showed that the answer sets of a nondisjunctive logic program are exactly the models of its Clark's completion that satisfy the loop formulas of all loops. Recently, Gebser and Schaub showed that the Lin-Zhao theorem remains correct even if we restrict loop formulas to a special class of loops called ``elementary loops.'' In this paper, we simplify and generalize the notion of an elementary loop, and clarify its role. We propose the notion of an elementary set, which is almost equivalent to the notion of an elementary loop for nondisjunctive programs, but is simpler, and, unlike elementary loops, can be extended to disjunctive programs without producing unintuitive results. We show that the maximal unfounded elementary sets for the ``relevant'' part of a program are exactly the minimal sets among the nonempty unfounded sets. We also present a graph-theoretic characterization of elementary sets for nondisjunctive programs, which is simpler than the one proposed in (Gebser & Schaub 2005). Unlike the case of nondisjunctive programs, we show that the problem of deciding an elementary set is coNP-complete for disjunctive programs.