On Elementary Loops of Logic Programs

TL;DR

This paper explores elementary loops in logic programs, extending their definition to disjunctive programs, analyzing properties, and introducing the class of HEF programs, which generalize HCF programs and impact stable model semantics.

Contribution

It reformulates elementary loops, extends them to disjunctive programs, and introduces HEF programs, broadening understanding of stable model semantics and loop recognition complexity.

Findings

Recognition of elementary loops is tractable for nondisjunctive programs.

Recognition is coNP-complete for disjunctive programs.

HEF programs generalize HCF programs and can be transformed into nondisjunctive programs efficiently.

Abstract

Using the notion of an elementary loop, Gebser and Schaub refined the theorem on loop formulas due to Lin and Zhao by considering loop formulas of elementary loops only. In this article, we reformulate their definition of an elementary loop, extend it to disjunctive programs, and study several properties of elementary loops, including how maximal elementary loops are related to minimal unfounded sets. The results provide useful insights into the stable model semantics in terms of elementary loops. For a nondisjunctive program, using a graph-theoretic characterization of an elementary loop, we show that the problem of recognizing an elementary loop is tractable. On the other hand, we show that the corresponding problem is {\sf coNP}-complete for a disjunctive program. Based on the notion of an elementary loop, we present the class of Head-Elementary-loop-Free (HEF) programs, which strictly generalizes the class of Head-Cycle-Free (HCF) programs due to Ben-Eliyahu and Dechter. Like an HCF program, an HEF program can be turned into an equivalent nondisjunctive program in polynomial time by shifting head atoms into the body.