## Logic Programs with Abstract Constraint Atoms (2004)

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Venue: | In Proceedings of the 19th National Conference on Artificial Intelligence (AAAI-04 |

Citations: | 23 - 5 self |

### BibTeX

@INPROCEEDINGS{Marek04logicprograms,

author = {Victor W. Marek and Miroslaw Truszczynski and Mirosl/aw Truszczy Nski},

title = {Logic Programs with Abstract Constraint Atoms},

booktitle = {In Proceedings of the 19th National Conference on Artificial Intelligence (AAAI-04},

year = {2004},

pages = {86--91},

publisher = {AAAI Press}

}

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### Abstract

We propose and study extensions of logic programming with constraints represented as generalized atoms of the form C(X), where X is a finite set of atoms and C is an abstract constraint (formally, a collection of sets of atoms). Atoms C(X) are satisfied by an interpretation (set of atoms) M , if M C. We focus here on monotone constraints, that is, those collections C that are closed under the superset. They include, in particular, weight (or pseudo-boolean) constraints studied both by the logic programming and SAT communities. We show that key concepts of the theory of normal logic programs such as the one-step provability operator, the semantics of supported and stable models, as well as several of their properties including complexity results, can be lifted to such case.

### Citations

546 |
Knowledge Representation, Reasoning and Declarative Problem Solving
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(Show Context)
Citation Context ... Lifschitz 2002; Lin & Zhao 2002), the importance of the formalism increased significantly as it became possible to use it not only as a modeling language but also as a practical computational tool. (=-=Baral 2003-=-) and (Gelfond & Copyright c○ 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. Leone 2002) provide a detailed discussion of the formalism and its application... |

343 | The DLV System for knowledge representation and reasoning
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Citation Context ...t action, diagnosis and abduction, product configuration, and modeling and reasoning about preferences. Moreover, due to the emergence of fast methods to compute stable models (Niemelä & Simons 1997; =-=Leone et al. 2003-=-; Babovich & Lifschitz 2002; Lin & Zhao 2002), the importance of the formalism increased significantly as it became possible to use it not only as a modeling language but also as a practical computati... |

320 | Extending and implementing the stable model semantics - Simons, Niemelä, et al. |

269 | Smodels – an implementation of the stable model and well-founded semantics for normal logic programs
- Niemelä, Simons
- 1997
(Show Context)
Citation Context ...lanning, reasoning about action, diagnosis and abduction, product configuration, and modeling and reasoning about preferences. Moreover, due to the emergence of fast methods to compute stable models (=-=Niemelä & Simons 1997-=-; Leone et al. 2003; Babovich & Lifschitz 2002; Lin & Zhao 2002), the importance of the formalism increased significantly as it became possible to use it not only as a modeling language but also as a ... |

211 | Assat: Computing answer set of a logic program by sat solvers
- Lin, Zhao
- 2004
(Show Context)
Citation Context ...figuration, and modeling and reasoning about preferences. Moreover, due to the emergence of fast methods to compute stable models (Niemelä & Simons 1997; Leone et al. 2003; Babovich & Lifschitz 2002; =-=Lin & Zhao 2002-=-), the importance of the formalism increased significantly as it became possible to use it not only as a modeling language but also as a practical computational tool. (Baral 2003) and (Gelfond & Copyr... |

138 | Well-founded semantics coincides with three-valued stable-semantics, Fund
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- 1990
(Show Context)
Citation Context ...he same way as the corresponding concepts are defined in normal logic programming in terms of the operator TP . In particular, the algebraic treatment of logic programming developed in (Fitting 2002; =-=Przymusinski 1990-=-; Denecker, Marek, & Truszczyński 2000) applies literally to deterministic F-programs. We note that this comment extends to 3- and 4-valued semantics of partial models, supported models and stable mod... |

110 | Fixpoint semantics for logic programming — A survey - Fitting |

101 |
The stable semantics for logic programs
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(Show Context)
Citation Context ...ides with the least Herbrand model of P . AUTOMATED REASONING 89Stable models of F-programs We will now define stable models of F-programs. To this end, we will generalize the concept of the reduct (=-=Gelfond & Lifschitz 1988-=-) to the case of F-programs and exploit results on Horn F-programs from the previous section. Definition 4 Let F be a class of monotone constraints and let P be an F-program. For a set of atoms M ⊆ At... |

85 | 2002, ‘PBS: A backtrack search pseudo Boolean solver’. In: Symposium on the theory and applications os satisfiability testing (SAT
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(Show Context)
Citation Context ..., they are also referred to as pseudo-boolean constraints. 2 For similar reasons, extending propositional logic with aggregates (most notably, weight constraints) has recently been gaining attention (=-=Aloul et al. 2002-=-; Dixon & Ginsberg 2002; Prestwich 2002). 86 AUTOMATED REASONINGjunctive) programs, but are limited in that they do not allow aggregate constructs to appear in the heads of the clauses. In addition, ... |

85 | Logic programming and knowledge representation – the A-Prolog perspective - Gelfond, Leone - 2002 |

62 | Strong equivalence made easy: Nested expressions and weight constraints. Theory and Practice of Logic Programming 3(4–5):609–622 - Turner |

53 |
Weight constraints as nested expressions. Theory and Practice of Logic Programming 5:45–74
- Ferraris, Lifschitz
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(Show Context)
Citation Context ...ely on the semantics of stable models. Furthermore, the connections to normal logic programming are indirect and somewhat obscure. There is a correspondence to logic programs with nested expressions (=-=Ferraris & Lifschitz 2004-=-), which helped establish some interesting results on programs with weight constraints (for instance, characterizations of their strong equivalence (Turner 2003)). However, the direct encodings of pro... |

48 | Ultimate well-founded and stable semantic for logic programs with aggregates - Denecker, Pelov, et al. - 2001 |

42 | Inference methods for a pseudo-boolean satisfiability solver
- Ginsberg
- 2002
(Show Context)
Citation Context ...erred to as pseudo-boolean constraints. 2 For similar reasons, extending propositional logic with aggregates (most notably, weight constraints) has recently been gaining attention (Aloul et al. 2002; =-=Dixon & Ginsberg 2002-=-; Prestwich 2002). 86 AUTOMATED REASONINGjunctive) programs, but are limited in that they do not allow aggregate constructs to appear in the heads of the clauses. In addition, all approaches discusse... |

41 | Aggregate functions in disjunctive logic programming: Semantics, complexity, and implementation in dlv - Dell’Armi, Faber, et al. - 2003 |

25 |
Logic Programming"; Handbook of Theoretical Computer Science
- Apt
- 1990
(Show Context)
Citation Context ... Emden & Kowalski 1976). Thanks to close parallels between these two concepts, we are able to reconstruct operator-based characterizations of models, supported models and stable models, presented in (=-=Apt 1990-=-; Fitting 2002). We also distinguish and discuss the class of deterministic programs (programs with clauses whose heads can be satisfied only in one way). For these programs the one-step provability o... |

23 | Randomised backtracking for linear pseudo-boolean constrainty problems - Prestwich - 2002 |

18 | Approximations, stable operators, well-founded fixpoints and applications in nonmonotonic reasoning - Denecker, Marek, et al. - 2000 |

18 | Stoodels: An implementation of the stable model and well-founded semantics for normal LP - Niemel', Simons - 1997 |

16 | Partial stable models for logic programs with aggregates - Pelov, Denecker, et al. - 2004 |

11 |
Negation as failure in Logic and Data Bases
- Clark
- 1978
(Show Context)
Citation Context ...bly, also some atoms that do not have such support. Models that contain only atoms with support form an important class of models generalizing the class of supported models for normal logic programs (=-=Clark 1978-=-; Apt 1990). Definition 2 Let F be a class of monotone constraints and let P be an F-program. A set of atoms M is a supported model of P if M is a model of P and M ⊆ hset(P (M)). Supported models have... |

7 | Characterizing stable models of logic programs with cardinality constraints - Marek, Niemelä, et al. - 2004 |

1 | Niemel a, I.; and Truszczy nski, M. 2004. Characterizing stable models of logic programs with cardinality constraints - Marek |