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On elementary loops of logic programs

Published online by Cambridge University Press:  24 May 2011

MARTIN GEBSER ,
JOOHYUNG LEE  and
YULIYA LIERLER
MARTIN GEBSER
Affiliation:
Institut für Informatik, Universität Potsdam, Germany (e-mail: gebser@cs.uni-potsdam.de)
JOOHYUNG LEE
Affiliation:
School of Computing, Informatics and Decision Systems Engineering, Arizona State University, Tempe, AZ, USA (e-mail: joolee@asu.edu)
YULIYA LIERLER
Affiliation:
Department of Computer Science, University of Kentucky, Lexington, KY, USA (e-mail: yuliya@cs.uky.edu)
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Abstract

Using the notion of an elementary loop, Gebser and Schaub (2005. Proceedings of the Eighth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'05), 53–65) refined the theorem on loop formulas attributable to Lin and Zhao (2004) by considering loop formulas of elementary loops only. In this paper, we reformulate the 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 also show that the corresponding problem is 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 attributable to Ben-Eliyahu and Dechter (1994. Annals of Mathematics and Artificial Intelligence 12, 53–87). 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.

Keywords

stable model semanticsloop formulasunfounded sets
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 11 , Issue 6 , November 2011 , pp. 953 - 988
Copyright
Copyright © Cambridge University Press 2011

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