### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 953-988 |

Number of pages | 36 |

Journal | Theory and Practice of Logic Programming |

Volume | 11 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2011 |

### Fingerprint

### Keywords

- loop formulas
- stable model semantics
- unfounded sets

### ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Hardware and Architecture
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Theory and Practice of Logic Programming*,

*11*(6), 953-988. https://doi.org/10.1017/S1471068411000019

**On elementary loops of logic programs.** / Gebser, Martin; Lee, Joohyung; Lierler, Yuliya.

Research output: Contribution to journal › Article

*Theory and Practice of Logic Programming*, vol. 11, no. 6, pp. 953-988. https://doi.org/10.1017/S1471068411000019

}

TY - JOUR

T1 - On elementary loops of logic programs

AU - Gebser, Martin

AU - Lee, Joohyung

AU - Lierler, Yuliya

PY - 2011/11

Y1 - 2011/11

N2 - 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.

AB - 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.

KW - loop formulas

KW - stable model semantics

KW - unfounded sets

UR - http://www.scopus.com/inward/record.url?scp=80155212946&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80155212946&partnerID=8YFLogxK

U2 - 10.1017/S1471068411000019

DO - 10.1017/S1471068411000019

M3 - Article

VL - 11

SP - 953

EP - 988

JO - Theory and Practice of Logic Programming

JF - Theory and Practice of Logic Programming

SN - 1471-0684

IS - 6

ER -