## Ultimate Well-founded and Stable Semantics for Logic Programs With Aggregates (Extended Abstract) (2001)

### Cached

### Download Links

- [www.cs.kuleuven.ac.be]
- [www.cs.kuleuven.ac.be]
- [www.cs.kuleuven.be]
- [www.cs.kuleuven.ac.be]
- [www.cs.kuleuven.be]
- [arxiv.org]
- [arxiv.org]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proceedings of ICLP-01, LNCS 2237 |

Citations: | 49 - 8 self |

### BibTeX

@INPROCEEDINGS{Denecker01ultimatewell-founded,

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

title = {Ultimate Well-founded and Stable Semantics for Logic Programs With Aggregates (Extended Abstract)},

booktitle = {In Proceedings of ICLP-01, LNCS 2237},

year = {2001},

pages = {212--226},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate well-founded model will be 2-valued and will coincide with the least fixpoint of TP . This is not the case for the standard well-founded semantics. For example in the standard well-founded model of the program: # p. p. p is undefined while the associated TP operator is monotone and p is true in the ultimate well-founded model. One disadvantage of using the ultimate semantics is that it has a higher computational cost even for programs without aggregates. The complexity goes one level higher in the polynomial hierarchy to # 2 for the well-founded model and to 2 for a stable model which is also complete for this class [2]. Fortunately, by adding aggregates the complexity does not increase further. To give an example of a logic program with aggregates we consider the problem of computing the length of the shortest path between two nodes in a direc

### Citations

1559 | The stable model semantics for logic programming
- Gelfond, Lifschitz
- 1988
(Show Context)
Citation Context ...tune the program in such a way that it satises the syntactic restrictions. For this reason, a number of attempts have been made for extending the well-founded semantics [21] and the stable semantics [=-=1-=-0] for unstratied programs with aggregates. Well-founded semantics is a natural semantics for deductive databases. As shown in [2, 4] it captures a general principle of non-monotone inductive denition... |

1176 |
Introduction to Lattices and Order
- Davey, Priestley
- 1990
(Show Context)
Citation Context ...lete lattice. The aggregate function glb: ℘(D) → D is antimonotone with respect to ≤ and the aggregate function lub: ℘(D) → D is monotone with respect to ≤. Proof Standard result from lattice theory (=-=Davey and Pristley 1990-=-, Lemma 2.22). In Table 1 we use the following notation for subsets of real numbers: R + for the set of non-negative numbers, R − for the set of non-positive numbers, R [1,∞) for the set of numbers in... |

879 | The well-founded semantics for general logic programs
- Gelder, Ross, et al.
- 1991
(Show Context)
Citation Context ...roblems, one has to carefully tune the program in such a way that it satises the syntactic restrictions. For this reason, a number of attempts have been made for extending the well-founded semantics [=-=21-=-] and the stable semantics [10] for unstratied programs with aggregates. Well-founded semantics is a natural semantics for deductive databases. As shown in [2, 4] it captures a general principle of no... |

735 | The semantics of predicate logic as a programming language - Emden, Kowalski - 1976 |

634 |
Towards a theory of declarative knowledge
- Apt, Blair, et al.
- 1988
(Show Context)
Citation Context ... The important class of stratified aggregate programs was considered already by Mumick et al. (1990) and Dell’Armi et al. (2003). It is a natural extension of the concept of stratified logic program (=-=Apt et al. 1988-=-) where aggregates are treated as negative literals. An aggregate program P is stratified if for each defined predicate p, there is a unique natural number ‖p‖ > 0 called the level of p such that if q... |

548 | Abductive Logic Programming
- Kakas, Kowalski, et al.
- 1993
(Show Context)
Citation Context ...ge representation which integrates classical logic assertions and inductive denitions. ID-logic can be seen as an extension and provides an epistemological foundation for Abductive logic Programming [=-=11]-=-. Also the stable semantics has been shown to be important for knowledge representation, in particular for nonmonotonic reasoning. Since recently, it is used as the foundation of an emerging paradigm ... |

393 | Fixedpoint semantics for logic programming
- Fitting
(Show Context)
Citation Context ...d exactly to the 3-valued interpretations (since I n J is empty). In terms of the bilattice representation of 4-valued interpretations, the fourvalued immediate consequence operator TP of a program P =-=[-=-9] can be dened as the operator that maps any pair (I ; J) to (T 1 P (I ; J); T 1 P (J; I)). Here, the operator T 1 P (I ; J) is dened as the 2-valued interpretation I 0 such that a ground atom p is t... |

334 | Logic programs with stable model semantics as a constraint programming paradigm
- Niemelä
- 1998
(Show Context)
Citation Context ...olving combinatorial search problems showed that many of these problems cannot be adequately expressed without aggregation. Recently, one of the prominent systems for stable logic programming smodels =-=[15]-=- has been extended successfully with a limited form of aggregate constraints [16]. [22] describes an extension of an abductive system with aggregates and some experiments with this system in the conte... |

322 | Extending and implementing the stable model semantics
- Simons, Niemelä, et al.
- 2002
(Show Context)
Citation Context ...variants of the set of and bag of procedures (Kemp and Stuckey 1991; Mumick et al. 1990). Recently, they were introduced in the context of two extensions of logic programming, Answer Set Programming (=-=Simons et al. 2002-=-) and Abductive Logic Programming (Van Nuffelen and Denecker 2000). Aggregates commonly show up in human expert knowledge and expressions of computational problems. For instance: • the query for the a... |

254 | Stable models and an alternative logic programming paradigm, in: The Logic Programming Paradigm: a 25-Year Perspective (Springer - Marek, Truszczyński - 1999 |

205 | Strongly equivalent logic programs
- Lifschitz, Pearce, et al.
(Show Context)
Citation Context ...emantics for such propositional formulas is different than the ˜ D-stable semantics which we defined. The main reason is that the semantics of (Ferraris 2005) is based on the logic of here-and-there (=-=Lifschitz et al. 2001-=-) while the ˜ D-stable semantics are based on Kleene’s strong three-valued logic T HREE. The simplest example of the difference between the two logics is that ¬¬p is equivalent to p in T HREE while it... |

96 | Recursive aggregates in disjunctive logic programs: Semantics and complexity - Faber, Leone, et al. - 2004 |

87 |
First order predicate logic with generalized quantifiers, Theoria 32
- Lindström
- 1966
(Show Context)
Citation Context ...itions is an important topic for future research but it is beyond the scope of this paper to investigate this issue. 8 Related Work Aggregate relations are closely related to generalized quantifiers (=-=Lindström 1966-=-). An example of a generalized quantifier is Most(A, B) which expresses that most elements of set A belong to set B. Clearly, this relation can be viewed as a binary aggregate relation with two set ar... |

73 | Answer sets for propositional theories - Ferraris |

66 | Semantics of Logic Programs with Aggregates
- Kemp, Stuckey
- 1991
(Show Context)
Citation Context ...sion of an abductive system with aggregates and some experiments with this system in the context of scheduling and puzzles. As will be shown in section 5, current extensions of well-founded semantics =-=[12, 18, 19]-=- and stable model semantics [12, 16] of programs with aggregates are still weak in some ways. In this paper, we propose a stronger extension of the well-founded and stable model semantics for logic pr... |

66 | Monotonic aggregation in deductive databases
- Ross, Sagiv
- 1992
(Show Context)
Citation Context ...well-known benchmark problems such as the shortest path program and the company control problem. 1 Introduction Programs with aggregates have been investigated in the context of database applications =-=[1, 14, 17, 20]-=-. One important approach to programs with aggregates is Monotone aggregation [17, 20]. This approach concentrates on programs where the 2-valued immediate consequence operator is monotone with respect... |

65 | Extending classical logic with inductive definitions
- Denecker
- 2000
(Show Context)
Citation Context ...ograms with aggregates. Well-founded semantics is a natural semantics for deductive databases. As shown in [2, 4] it captures a general principle of non-monotone inductive denition. For this reason, [=-=3-=-] used the well-founded semantics as the semantic principle of ID-logic, a logic for knowledge representation which integrates classical logic assertions and inductive denitions. ID-logic can be seen ... |

62 | The magic of duplicates and aggregates
- Pirahesh, Mumick, et al.
- 1990
(Show Context)
Citation Context ...well-known benchmark problems such as the shortest path program and the company control problem. 1 Introduction Programs with aggregates have been investigated in the context of database applications =-=[1, 14, 17, 20]-=-. One important approach to programs with aggregates is Monotone aggregation [17, 20]. This approach concentrates on programs where the 2-valued immediate consequence operator is monotone with respect... |

60 | Representing knowledge in a-prolog - Gelfond |

52 | Stable model semantics of weight constraint rules
- Niemelä, Simons, et al.
- 1999
(Show Context)
Citation Context ... adequately expressed without aggregation. Recently, one of the prominent systems for stable logic programming smodels [15] has been extended successfully with a limited form of aggregate constraints =-=[16]-=-. [22] describes an extension of an abductive system with aggregates and some experiments with this system in the context of scheduling and puzzles. As will be shown in section 5, current extensions o... |

45 | The well-founded semantics is the principle of inductive definition
- Denecker
- 1998
(Show Context)
Citation Context ...or extending the well-founded semantics [21] and the stable semantics [10] for unstratied programs with aggregates. Well-founded semantics is a natural semantics for deductive databases. As shown in [=-=2, 4-=-] it captures a general principle of non-monotone inductive denition. For this reason, [3] used the well-founded semantics as the semantic principle of ID-logic, a logic for knowledge representation w... |

43 | Uniform semantic treatment of default and autoepistemic logics
- Denecker, Marek, et al.
- 2003
(Show Context)
Citation Context ...)sxpoints of O as the stable (respectively Kripke-Kleene and well-founded)sxpoints of its ultimate partial approximation U lt(O). 3.1 Approximations of Logic Programs Approximation theory was used in =-=[7]-=- for describing the semantics of Logic Programming, Default logic and Autoepistemic logic. In the case of Logic Programming, the underlying lattice is the lattice of Herbrand interpretations. The bila... |

41 |
The well-founded semantics of aggregation
- Gelder
- 1992
(Show Context)
Citation Context ...sion of an abductive system with aggregates and some experiments with this system in the context of scheduling and puzzles. As will be shown in section 5, current extensions of well-founded semantics =-=[12, 18, 19]-=- and stable model semantics [12, 16] of programs with aggregates are still weak in some ways. In this paper, we propose a stronger extension of the well-founded and stable model semantics for logic pr... |

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

37 | Logic programming revisited: logic programs as inductive definitions
- Denecker, Bruynooghe, et al.
- 2001
(Show Context)
Citation Context ...or extending the well-founded semantics [21] and the stable semantics [10] for unstratied programs with aggregates. Well-founded semantics is a natural semantics for deductive databases. As shown in [=-=2, 4-=-] it captures a general principle of non-monotone inductive denition. For this reason, [3] used the well-founded semantics as the semantic principle of ID-logic, a logic for knowledge representation w... |

36 | A framework for representing and solving NP search problems
- Mitchell, Ternovska
- 2005
(Show Context)
Citation Context ...el generation for aggregate programs may remain tractable under an appropriate choice of the three valued aggregates. The type of computational problem considered here is the model extension problem (=-=Mitchell and Ternovska 2005-=-): given a signature Σ(Π), an aggregate program P based on Σ(Π) and an input Σ-structure ˜ D which is two-valued on all predicates and three-valued on aggregate symbols, compute the Kripke-Kleene mode... |

28 |
A constructive semantic characterization of aggregates in answer set programming
- Son, Pontelli
- 2007
(Show Context)
Citation Context ... not satisfied by the semantics defined in (Faber et al. 2004). This is a fundamental weakness of this semantics. Finally, we mention another recent work on defining a stable semantics of aggregates (=-=Son and Pontelli 2007-=-). As shown in that paper, the semantics defined there coincides with the ult-stable semantics and provides an interesting different formalization for this semantics. 9 Conclusions and Future Work In ... |

26 |
Approximating operators, stable operators, well-founded fixpoints and applications in non-monotonic reasoning
- Denecker, Marek, et al.
- 2000
(Show Context)
Citation Context ... Celestijnenlaan 200A, 3001 Heverlee, Belgium In [3] we investigate the problem of defining a well-founded and stable semantics for programs with aggregates. Our work is based on Approximation Theorys=-=[1]-=- which is a general algebraic framework for approximating non-monotone operators on a complete lattice L by approximating operators on the bi-lattice L 2 . The theory identifies basic properties of th... |

26 | Semantics of logic programs with aggregates - Pelov |

25 | Foundations of aggregations in deductive databases
- Gelder
- 1993
(Show Context)
Citation Context ...well-known benchmark problems such as the shortest path program and the company control problem. 1 Introduction Programs with aggregates have been investigated in the context of database applications =-=[1, 14, 17, 20]-=-. One important approach to programs with aggregates is Monotone aggregation [17, 20]. This approach concentrates on programs where the 2-valued immediate consequence operator is monotone with respect... |

24 | Logic programs with abstract constraint atoms
- Marek, Truszczyński
- 2004
(Show Context)
Citation Context ...the rules. This approach have been further developed in different directions. One line of research was to consider different variations and extensions of weight constraints like abstract constraints (=-=Marek and Truszczyński 2004-=-), monotone cardinality atoms (Marek et al. 2004) or set constraints (Marek and Remmel 2004). Such constraint atoms correspond in a natural way to aggregate atoms. The stable semantics of these extens... |

18 | Extending the Well-Founded and Valid Semantics for Aggregation
- Sudarshan, Srivastava, et al.
- 1993
(Show Context)
Citation Context ...sion of an abductive system with aggregates and some experiments with this system in the context of scheduling and puzzles. As will be shown in section 5, current extensions of well-founded semantics =-=[12, 18, 19]-=- and stable model semantics [12, 16] of programs with aggregates are still weak in some ways. In this paper, we propose a stronger extension of the well-founded and stable model semantics for logic pr... |

16 | Set Constraints in Logic Programming - Marek, Remmel - 2004 |

13 |
M.: Ultimate Well-Founded and Stable Model Semantics for Logic Programs with Aggregates
- Denecker, Pelov, et al.
- 2001
(Show Context)
Citation Context ...ates Marc Denecker a Nikolay Pelov b Maurice Bruynooghe b a Universite Libre de Bruxelles, Bld du Triomphe CP 212, 1050 Brussels, Belgium b K.U.Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium In =-=[3]-=- we investigate the problem of defining a well-founded and stable semantics for programs with aggregates. Our work is based on Approximation Theorys[1] which is a general algebraic framework for appro... |

13 | Ultimate approximation and its application in nonmonotonic knowledge representation systems
- Denecker, Marek, et al.
- 2001
(Show Context)
Citation Context ...le semantics (Gelfond and Lifschitz 1988) and the well-founded semantics (Van Gelder et al. 1991). The foundation of our work is the algebraic theory of approximating operators (Denecker et al. 2000; =-=Denecker et al. 2004-=-). Approximation theory is a fixpoint theory of non-monotone lattice operators. With any lattice operator O: L → L, it associates a family of approximating operators A: L 2 → L 2 on the product lattic... |

10 | Ultimate approximations in nonmonotonic knowledge representation systems
- Denecker, Marek, et al.
- 2002
(Show Context)
Citation Context ...gramming. For example the WF fixpoint is more precise than the KK fixpoint and also approximates every stable fixpoint. Moreover, stable fixpoints are always minimal fixpoints. In a more recent paper =-=[2]-=- the authors study the concept of precision of an approximating operator. They show that more precise approximating operators have more precise KK and WF fixpoints and a larger number of stable fixpoi... |

8 |
Partial stable semantics for logic programs with aggregates
- Pelov, Denecker, et al.
- 2004
(Show Context)
Citation Context ...ecursion over aggregation, for example the Company Controls program from Example 5.2 (Pelov 2004, Section 4.3.3). For aggregate programs containing only negative aggregate literals it has been shown (=-=Pelov et al. 2004-=-) that the set of ult-stables models is the same as the set of stable models defined by the above authors. A more elaborate definition of a stable semantics was given by (Simons et al. 2002) for progr... |

7 |
M.: On Well-Behaved Semantics Suitable for Aggregation
- Dix, Osorio
- 1997
(Show Context)
Citation Context ...ics of programs with aggregates one can still work with aggregate functions dened on standard sets. Further investigation is needed about the precise relation of these two approaches. Dix and Osorio [=-=8]-=- have also observed that the standard well-founded semantics is not strong enough for programs with aggregation. They have pointed out that the WFS + (introduced independently by Dix and Schlipf) is w... |

7 | Logic programs with monotone cardinality atoms
- Marek, Niemelä, et al.
- 2004
(Show Context)
Citation Context ...fferent directions. One line of research was to consider different variations and extensions of weight constraints like abstract constraints (Marek and Truszczyński 2004), monotone cardinality atoms (=-=Marek et al. 2004-=-) or set constraints (Marek and Remmel 2004). Such constraint atoms correspond in a natural way to aggregate atoms. The stable semantics of these extensions is also defined in terms of lattice operato... |

5 |
Problem solving in id-logic with aggregates: Some experiments
- Nuelen, Denecker
- 2000
(Show Context)
Citation Context ...ately expressed without aggregation. Recently, one of the prominent systems for stable logic programming smodels [15] has been extended successfully with a limited form of aggregate constraints [16]. =-=[22]-=- describes an extension of an abductive system with aggregates and some experiments with this system in the context of scheduling and puzzles. As will be shown in section 5, current extensions of well... |

3 | Quantifiers and partiallity - Eijck - 1996 |

2 |
M.: Ultimate approximations
- Denecker, Marek, et al.
- 2001
(Show Context)
Citation Context ... Both are minimal, but clearly only the second is intended. Thesrstsxpoint is not acceptable becausesp(0) only depends positively on itself, hence, the model is not well-founded. Approximation theory =-=[5, 6]-=- is asxpoint theory that assignssxpoints to any nonmonotone operator in a complete lattice and was used to describe the semantics of logic programs, default logic and autoepistemic logic. The next sec... |

2 | Problem solving in ID-logic with aggregates: some experiments - Nuffelen, Denecker - 2000 |

1 | First order predicate logic with generalized quantifiers - Saraswat, Ueda, et al. - 1966 |

1 |
Smodels with CLP and its applica- Nikolay Pelov, Marc Denecker, Maurice Bruynooghe tions: A simple and effective approach to aggregates in ASP
- Elkabani, Pontelli, et al.
- 2004
(Show Context)
Citation Context ...le semantics of general logic programs with negation to aggregate programs was to treat aggregate literals as negative literals when computing the program reduct (Kemp and Stuckey 1991; Gelfond 2002; =-=Elkabani et al. 2004-=-). Such semantics is relatively easy to define and the definition also extends to answer sets of disjunctive logic programs. However, it has been46 Nikolay Pelov, Marc Denecker, Maurice Bruynooghe sh... |