### Abstract

By introducing the concepts of a loop and a loop formula, Lin and Zhao showed that the answer sets of a nondisjunctive logic program are exactly the models of its Clark's completion that satisfy the loop formulas of all loops. Recently, Gebser and Schaub showed that the Lin-Zhao theorem remains correct even if we restrict loop formulas to a special class of loops called "elementary loops." In this paper, we simplify and generalize the notion of an elementary loop, and clarify its role. We propose the notion of an elementary set, which is almost equivalent to the notion of an elementary loop for nondisjunctive programs, but is simpler, and, unlike elementary loops, can be extended to disjunctive programs without producing unintuitive results. We show that the maximal unfounded elementary sets for the "relevant" part of a program are exactly the minimal sets among the nonempty unfounded sets. We also present a graph-theoretic characterization of elementary sets for nondisjunctive programs, which is simpler than the one proposed in (Gebser & Schaub 2005). Unlike the case of nondisjunctive programs, we show that the problem of deciding an elementary set is coNP-complete for disjunctive programs.

Original language | English (US) |
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Title of host publication | Proceedings of the National Conference on Artificial Intelligence |

Pages | 244-249 |

Number of pages | 6 |

Volume | 1 |

State | Published - 2006 |

Event | 21st National Conference on Artificial Intelligence and the 18th Innovative Applications of Artificial Intelligence Conference, AAAI-06/IAAI-06 - Boston, MA, United States Duration: Jul 16 2006 → Jul 20 2006 |

### Other

Other | 21st National Conference on Artificial Intelligence and the 18th Innovative Applications of Artificial Intelligence Conference, AAAI-06/IAAI-06 |
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Country | United States |

City | Boston, MA |

Period | 7/16/06 → 7/20/06 |

### ASJC Scopus subject areas

- Software

### Cite this

*Proceedings of the National Conference on Artificial Intelligence*(Vol. 1, pp. 244-249)

**Elementary sets for logic programs.** / Gebser, Martin; Lee, Joohyung; Lierler, Yuliya.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the National Conference on Artificial Intelligence.*vol. 1, pp. 244-249, 21st National Conference on Artificial Intelligence and the 18th Innovative Applications of Artificial Intelligence Conference, AAAI-06/IAAI-06, Boston, MA, United States, 7/16/06.

}

TY - GEN

T1 - Elementary sets for logic programs

AU - Gebser, Martin

AU - Lee, Joohyung

AU - Lierler, Yuliya

PY - 2006

Y1 - 2006

N2 - By introducing the concepts of a loop and a loop formula, Lin and Zhao showed that the answer sets of a nondisjunctive logic program are exactly the models of its Clark's completion that satisfy the loop formulas of all loops. Recently, Gebser and Schaub showed that the Lin-Zhao theorem remains correct even if we restrict loop formulas to a special class of loops called "elementary loops." In this paper, we simplify and generalize the notion of an elementary loop, and clarify its role. We propose the notion of an elementary set, which is almost equivalent to the notion of an elementary loop for nondisjunctive programs, but is simpler, and, unlike elementary loops, can be extended to disjunctive programs without producing unintuitive results. We show that the maximal unfounded elementary sets for the "relevant" part of a program are exactly the minimal sets among the nonempty unfounded sets. We also present a graph-theoretic characterization of elementary sets for nondisjunctive programs, which is simpler than the one proposed in (Gebser & Schaub 2005). Unlike the case of nondisjunctive programs, we show that the problem of deciding an elementary set is coNP-complete for disjunctive programs.

AB - By introducing the concepts of a loop and a loop formula, Lin and Zhao showed that the answer sets of a nondisjunctive logic program are exactly the models of its Clark's completion that satisfy the loop formulas of all loops. Recently, Gebser and Schaub showed that the Lin-Zhao theorem remains correct even if we restrict loop formulas to a special class of loops called "elementary loops." In this paper, we simplify and generalize the notion of an elementary loop, and clarify its role. We propose the notion of an elementary set, which is almost equivalent to the notion of an elementary loop for nondisjunctive programs, but is simpler, and, unlike elementary loops, can be extended to disjunctive programs without producing unintuitive results. We show that the maximal unfounded elementary sets for the "relevant" part of a program are exactly the minimal sets among the nonempty unfounded sets. We also present a graph-theoretic characterization of elementary sets for nondisjunctive programs, which is simpler than the one proposed in (Gebser & Schaub 2005). Unlike the case of nondisjunctive programs, we show that the problem of deciding an elementary set is coNP-complete for disjunctive programs.

UR - http://www.scopus.com/inward/record.url?scp=33750709265&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750709265&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33750709265

SN - 1577352815

SN - 9781577352815

VL - 1

SP - 244

EP - 249

BT - Proceedings of the National Conference on Artificial Intelligence

ER -