## Partial Stable Models for Logic Programs with Aggregates (2004)

Venue: | In: LPNMR-7. LNCS 2923 |

Citations: | 16 - 0 self |

### BibTeX

@INPROCEEDINGS{Pelov04partialstable,

author = {Nikolay Pelov and Marc Denecker and Maurice Bruynooghe},

title = {Partial Stable Models for Logic Programs with Aggregates},

booktitle = {In: LPNMR-7. LNCS 2923},

year = {2004},

pages = {207--219},

publisher = {Springer}

}

### OpenURL

### Abstract

We introduce a family of partial stable model semantics for logic programs with arbitrary aggregate relations. The semantics are parametrized by the interpretation of aggregate relations in three-valued logic. Any semantics in this family satisfies two important properties: (i) it extends the partial stable semantics for normal logic programs and (ii) total stable models are always minimal. We also give a specific instance of the semantics and show that it has several attractive features.

### Citations

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Citation Context ...d by transfinite iteration of S starting from the bottom element in the # p order which is (#, #). The standard partial approximating operator of the TP operator is Fitting's three-valued #P operator =-=[6]-=-. The Kripke-Kleene fixpoint of #P is equal to the Kripke-Kleene semantics of P [6]. Although partial stable models [14] are defined in a very di#erent way they do coincide with partial stable fixpoin... |

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Citation Context ...s giving precise relationship between our semantics and most of the previous proposals for stable model semantics for aggregate programs. This includes the stable semantics of weight constraint rules =-=[16]-=- used by the smodels system, the stable semantics of Kemp and Stuckey [11] which is also used by A-Prolog [8] and the dlv system [2], and our previous work on the ultimate semantics of aggregate progr... |

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Citation Context ...ints. In logic programming we are interested in approximating the fixpoints of the TP operator. One possible approximating operator of TP is the standard four-valued immediate consequence operator #P =-=[7]-=-. The partial stable fixpoints of #P , as defined by Approximation Theory [3], correspond with partial stable models as defined by Przymusinski [14] and Fitting [7]. Recently, Denecker et al. [4] have... |

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Citation Context ...us proposals for stable model semantics for aggregate programs. This includes the stable semantics of weight constraint rules [16] used by the smodels system, the stable semantics of Kemp and Stuckey =-=[11]-=- which is also used by A-Prolog [8] and the dlv system [2], and our previous work on the ultimate semantics of aggregate programs [5]. The structure of the paper is as follows. We start by defining th... |

66 | Monotonic aggregation in deductive databases
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Citation Context ... several attractive features. 1 Introduction Aggregates are an important concept for natural modeling of many problems. Existing work already covers a large class of aggregate programs, like monotone =-=[13, 15]-=- and stratified [1, 13] ones. There are, however, programs which involve recursion over non-monotone aggregation and do not fall in any of these classes. An example is the Party Invitation problem (Ex... |

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Citation Context ... several attractive features. 1 Introduction Aggregates are an important concept for natural modeling of many problems. Existing work already covers a large class of aggregate programs, like monotone =-=[13, 15]-=- and stratified [1, 13] ones. There are, however, programs which involve recursion over non-monotone aggregation and do not fall in any of these classes. An example is the Party Invitation problem (Ex... |

60 | Representing knowledge in a-prolog
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Citation Context ...ics for aggregate programs. This includes the stable semantics of weight constraint rules [16] used by the smodels system, the stable semantics of Kemp and Stuckey [11] which is also used by A-Prolog =-=[8]-=- and the dlv system [2], and our previous work on the ultimate semantics of aggregate programs [5]. The structure of the paper is as follows. We start by defining the syntax and semantics of aggregate... |

41 | Aggregate functions in disjunctive logic programming: semantics, complexity, and implementation in DLV
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Citation Context ...ures. 1 Introduction Aggregates are an important concept for natural modeling of many problems. Existing work already covers a large class of aggregate programs, like monotone [13, 15] and stratified =-=[1, 13]-=- ones. There are, however, programs which involve recursion over non-monotone aggregation and do not fall in any of these classes. An example is the Party Invitation problem (Example 1). The developme... |

41 | The well-founded semantics of aggregation - Gelder - 1992 |

27 |
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Citation Context ...tial stable models. Thus, our definition also covers these two semantics. The foundation of this work is the algebraic theory of approximating operators developed by Denecker, Marek, and Truszczynski =-=[3, 4]-=-. The theory studies approximations of the fixpoints of non-monotone lattice operators O : L # L. With any such operator O, it associates a family of approximating operators A : L 2 # L 2 on the produ... |

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Citation Context ...computation of the well-founded fixpoint based on the partial stable operator of #P is very similar to the bottom-up evaluation technique based on the doubled program by Kemp, Srivastava, and Stuckey =-=[10]-=-. 3.2 Partial Stable Models of Aggregate Programs We are now ready to define our semantics. Our goal is to define a partial approximating operator # aggr P of T aggr P which, for programs without aggr... |

14 |
Logic programs with cardinality constraints
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(Show Context)
Citation Context ...ow aggregates in the heads of rules. We think that this is not a serious limitation because such programs can be translated to programs without aggregates in the heads by introducing additional atoms =-=[12]-=-. For simplicity of the presentation, we defined the semantics for a propositional language and set expressions of finite size. It can be easily extended to programs with variables by considering a su... |

13 |
M.: Ultimate Well-Founded and Stable Model Semantics for Logic Programs with Aggregates
- Denecker, Pelov, et al.
- 2001
(Show Context)
Citation Context ...ggregate relation A r for every aggregate relation r. We use the idea of ultimate approximations [4]. In our previous work, we considered the ultimate approximating operator of the entire TP operator =-=[5]-=- while here we consider the ultimate approximation only of aggregate relations. Definition 8. Let r # M(D)sD be an aggregate relation. The ultimate approximating aggregate U r : M(D) c D # T HREE of r... |

11 | Ultimate approximations in nonmonotonic knowledge representation systems
- Denecker, Marek, et al.
- 2002
(Show Context)
Citation Context ...tial stable models. Thus, our definition also covers these two semantics. The foundation of this work is the algebraic theory of approximating operators developed by Denecker, Marek, and Truszczynski =-=[3, 4]-=-. The theory studies approximations of the fixpoints of non-monotone lattice operators O : L # L. With any such operator O, it associates a family of approximating operators A : L 2 # L 2 on the produ... |

5 | Aggregate functions in DLV
- Dell’Armi, Faber, et al.
- 2003
(Show Context)
Citation Context ...ams. This includes the stable semantics of weight constraint rules [16] used by the smodels system, the stable semantics of Kemp and Stuckey [11] which is also used by A-Prolog [8] and the dlv system =-=[2]-=-, and our previous work on the ultimate semantics of aggregate programs [5]. The structure of the paper is as follows. We start by defining the syntax and semantics of aggregate programs (Section 2). ... |

2 |
The well-founded semantics coincides with the three-valued stable semantics
- Przymusinksi
- 1990
(Show Context)
Citation Context ...andard four-valued immediate consequence operator #P [7]. The partial stable fixpoints of #P , as defined by Approximation Theory [3], correspond with partial stable models as defined by Przymusinski =-=[14]-=- and Fitting [7]. Recently, Denecker et al. [4] have investigated the semantics determined by another approximating operator of TP --- the ultimate approximating operator UP . This is the most precise... |