On Elementary Loops of Logic Programs
  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.
