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On elementary loops of logic programs

  • 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 generalizesUsing 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 Ha: program, an HEF program can be turned into an equivalent nondisjunctive program in polynomial time by shifting head atoms into the body.show moreshow less

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Metadaten
Author:Martin GebserORCiD, Joohyung Lee, Yuliya Lierler
DOI:https://doi.org/10.1017/S1471068411000019
ISSN:1471-0684 (print)
Parent Title (English):Theory and practice of logic programming
Publisher:Cambridge Univ. Press
Place of publication:New York
Document Type:Article
Language:English
Year of first Publication:2011
Year of Completion:2011
Release Date:2017/03/26
Tag:loop formulas; stable model semantics; unfounded sets
Volume:11
Issue:2
Pagenumber:36
First Page:953
Last Page:988
Funder:German Research Foundation [SCHA 550/8-1]; National Science Foundation [IIS-0916116, IIS-0712113]; Office of the Director of National Intelligence (ODNI); Intelligence Advanced Research Projects Activity (IARPA), through US army; Computing Innovation Fellowship
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Informatik und Computational Science
Peer Review:Referiert
Institution name at the time of publication:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Informatik
Notes extern:Zweitveröffentlichung in der Schriftenreihe Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 566