## Embedding non-ground logic programs into autoepistemic logic for knowledge base combination (2007)

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Venue: | IN PROC. IJCAI-2007 |

Citations: | 23 - 10 self |

### BibTeX

@INPROCEEDINGS{Bruijn07embeddingnon-ground,

author = {Jos De Bruijn and Thomas Eiter and Technische Universität Wien and Axel Polleres},

title = {Embedding non-ground logic programs into autoepistemic logic for knowledge base combination},

booktitle = {IN PROC. IJCAI-2007},

year = {2007},

pages = {304--309},

publisher = {AAAI Press/IJCAI}

}

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### Abstract

In the context of the Semantic Web, several approaches to the combination of ontologies, given in terms of theories of classical first-order logic and rule bases, have been proposed. They either cast rules into classical logic or limit the interaction between rules and ontologies. Autoepistemic logic (AEL) is an attractive formalism which allows to overcome these limitations, by serving as a uniform host language to embed ontologies and nonmonotonic logic programs into it. For the latter, so far only the propositional setting has been considered. In this paper, we present three embeddings of normal and three embeddings of disjunctive non-ground logic programs under the stable model semantics into first-order AEL. While all embeddings correspond with respect to objective ground atoms, differences arise when considering non-atomic formulas and combinations with first-order theories. We compare the embeddings with respect to stable expansions and autoepistemic consequences, considering the embeddings by themselves, as well as combinations with classical theories. Our results reveal differences and correspondences of the embeddings and

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Citation Context ...; Marek and Truszczyński 1993], is an attractive candidate for serving as a uniform host formalism for combinations. Compared to other well-known nonmonotonic formalisms, like Reiter’s default logic [=-=Reiter 1980-=-], FO-AEL offers a uniform language in which (nonmonotonic) rules themselves can be expressed at the object level. This conforms with the idea of treating an ontology and a logic program together as a... |

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Citation Context ...p, ¬Lq, . . . } and T EH 2 = {q, Lp, ¬Lp, . . . }; the latter include neither ¬q nor ¬p. Note that the embedding τHP cannot be naively extended to logic programs with strong (“classical”) negation ∼ [=-=Gelfond and Lifschitz 1991-=-], even for the propositional case. Take, for example, the logic program P = {p ←∼ p}; it has one stable model, namely M = ∅. The naive extension of τHP treats strong negation as negation in classical... |

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Citation Context ...etween three kinds of combinations of rules and ontologies: (1) uniform combinations (e.g., CARIN [Levy and Rousset 1998] and SWRL [Horrocks et al. 2005]), (2) hybrid combinations (e.g., dl-programs [=-=Eiter et al. 2008-=-] and DL+log [Rosati 2006]), and (3) embedding combinations (e.g., the MKNF combination by Motik and Rosati [2007] and a combination based on quantified equilibrium logic [de Bruijn et al. 2007]); for... |

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Citation Context ...Lifschitz 1994] and the subsequent embedding of MBNF into standard AEL. Finally, Chen also arrived at the EH embedding via MBNF, but he subsequently embedded MBNF in Levesque’s logic of only knowing [=-=Levesque 1990-=-], a subset of which corresponds with standard AEL. ACM Transactions on Computational Logic, Vol. V, No. N, Month 20YY.12 · Jos de Bruijn et al. Definition 4.1. Let r be a normal rule of the form (1)... |

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Citation Context ...lassical theory in AEL is trivial, and several embeddings of logic programs in AEL have been described [Gelfond and Lifschitz 1988; Marek and Truszczynski 1993; Lifschitz and Schwarz 1993; Chen 1993; =-=Przymusinski 1991-=-a]. However, they all have been developed for the propositional case only, whereas we need to deal with non-ground theories and programs. This requires us to consider first-order autoepistemic logic (... |

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Citation Context ...oited to derive properties. Investigating this in detail remains for future work. Several nonmonotonic extensions of description logics have been defined in the literature [Baader and Hollunder 1995; =-=Donini et al. 1998-=-; Donini et al. 2002; Bonatti et al. 2006]. These might be further extended to accommodate logic programs by well-known correspondences of the latter to nonmonotonic formalisms. In more detail, extens... |

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Citation Context ...ich we consider next. 9.2 Combinations of Rules and Ontologies Roughly speaking, we can distinguish between three kinds of combinations of rules and ontologies: (1) uniform combinations (e.g., CARIN [=-=Levy and Rousset 1998-=-] and SWRL [Horrocks et al. 2005]), (2) hybrid combinations (e.g., dl-programs [Eiter et al. 2008] and DL+log [Rosati 2006]), and (3) embedding combinations (e.g., the MKNF combination by Motik and Ro... |

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Citation Context ...s of this paper can be exploited to derive properties. Investigating this in detail remains for future work. Several nonmonotonic extensions of description logics have been defined in the literature [=-=Baader and Hollunder 1995-=-; Donini et al. 1998; Donini et al. 2002; Bonatti et al. 2006]. These might be further extended to accommodate logic programs by well-known correspondences of the latter to nonmonotonic formalisms. In... |

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Citation Context ...for logic programs. Adding classical negation to the τEB and τEH is routine, and has been done by de Bruijn et al. [2008] for FO-AEL with standard names. Other interesting extensions include nesting [=-=Lifschitz et al. 1999-=-], where the closeness between nesting in logic programs and the logic MKNF suggests that an embedding is straightforward, and aggregates [Faber et al. 2004; Ferraris 2005; Pelov et al. 2007; Son and ... |

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Citation Context ...nt subtle issues of quantification in FO-AEL. In more detail, we present three embeddings, τHP, τEB, and τEH , for normal logic programs which extend respective embeddings for the propositional case [=-=Gelfond 1987-=-; Gelfond and Lifschitz 1988; Marek and Truszczynski 1993; Chen 1993; Lifschitz 1994], and three embeddings, τ ∨ HP , τ ∨ EB , and τ ∨ EH , for disjunctive logic programs, where τ ∨ HP and τ ∨ EH exte... |

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Citation Context ...ion, including Horn, universal, and generalized Horn theories. The latter are of particular interest for ontologies, since they essentially include RDF Schema [de Bruijn and Heymans 2007], Horn-SHIQ [=-=Hustadt et al. 2005-=-], and the OWL 2 profiles QL, RL, and EL [OWL 2 Profiles 2009]; furthermore, they essentially include also Tuple Generating Dependencies [Abiteboul et al. 1995], which are a popular class of constrain... |

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Citation Context ...re, Marek and Truszczynski arrived at their embeddings through embeddings of logic programs in reflexive autoepistemic logic [Schwarz 1992], which is equivalent to McDermott’s nonmonotonic modal sw5 [=-=McDermott 1982-=-], and the subsequent embedding of reflexive autoepistemic logic into standard AEL. Lifschitz and Schwarz arrived at the EH embedding through an embedding of logic programs in Lifschitz’s nonmonotonic... |

91 | Recursive Aggregates in Disjunctive Logic Programs: Semantics and Complexity - Faber, Leone, et al. - 2004 |

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Citation Context ...erties. Investigating this in detail remains for future work. Several nonmonotonic extensions of description logics have been defined in the literature [Baader and Hollunder 1995; Donini et al. 1998; =-=Donini et al. 2002-=-; Bonatti et al. 2006]. These might be further extended to accommodate logic programs by well-known correspondences of the latter to nonmonotonic formalisms. In more detail, extensions of DL semantics... |

87 | The Logic of Knowledge Bases
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Citation Context ...he propositional case only, whereas we need to deal with non-ground theories and programs. This requires us to consider first-order autoepistemic logic (FO-AEL) [Konolige 1991; Kaminski and Rey 2002; =-=Levesque and Lakemeyer 2000-=-], and non-ground versions of these embeddings. We consider the semantics for FO-AEL as defined by Konolige [1991], because it faithfully extends first-order logic with equality (other variants are di... |

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Citation Context ...wn that reasoning in standard first-order logic can be reduced to reasoning in first-order logic with the standard names assumption, as long as there are sufficiently many constant symbols available [=-=Fitting 1996-=-]. Different from Levesque [1990], Kaminski and Rey [2002] did not consider equality in the language. However, equality in first-order logic with standard names behaves quite differently from equality... |

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Citation Context ...d by de Bruijn et al. [2008], who show how well-known approaches to combining rules and ontologies in the Semantic Web context can be embedded into FO-AEL, including [Eiter et al. 2008; Rosati 2006b; =-=Motik and Rosati 2007-=-]. Notably, the DL+log approach can be embedded into FO-AEL by adding further axioms to the simple combination that we have considered here. The companion paper confirms the value of our results, on w... |

64 |
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Citation Context ...behave under knowledge combination? Concerning the first question, Motik and Rosati [2007] use a variant of Lifschitz’s bimodal nonmonotonic logic of minimal knowledge and negation-as-failure (MKNF) [=-=Lifschitz 1991-=-]. While the proposed embeddings of the first-order (FO) theory and the logic program are both faithful in the sense described above, the particular combination proposed by Motik and Rosati is only on... |

61 | A Uniform Integration of Higher-order Reasoning and External Evaluations in Answer-set Programming
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Citation Context ...tology for a query. Eiter et al. [2008] show that query answering in dlprograms is decidable as long as reasoning in the individual components (ontology and logic program) is decidable. HEX-programs [=-=Eiter et al. 2005-=-] generalize dlprograms to more general external evaluations that are not limited to queries on DL ontologies. DL+log makes a distinction between ontology and rules predicates; rules predicates may no... |

57 | Wellfounded semantics for description logic programs in the semantic web - Eiter, Lukasiewicz, et al. - 2011 |

56 |
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Citation Context ...oited to derive properties. Investigating this in detail remains for future work. Several nonmonotonic extensions of description logics have been defined in the literature [Baader and Hollunder 1995; =-=Donini et al. 1998-=-; Donini et al. 2002; Bonatti et al. 2006]. These might be further extended to accommodate logic programs by well-known correspondences of the latter to nonmonotonic formalisms. In more detail, extens... |

52 | From SPARQL to rules (and back - Polleres - 2007 |

48 | OWL rules: A proposal and prototype implementation
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Citation Context ...tions of Rules and Ontologies Roughly speaking, we can distinguish between three kinds of combinations of rules and ontologies: (1) uniform combinations (e.g., CARIN [Levy and Rousset 1998] and SWRL [=-=Horrocks et al. 2005-=-]), (2) hybrid combinations (e.g., dl-programs [Eiter et al. 2008] and DL+log [Rosati 2006]), and (3) embedding combinations (e.g., the MKNF combination by Motik and Rosati [2007] and a combination ba... |

44 | Well-founded and stable semantics of logic programs with aggregates. Theory and Practice of Logic Programming
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Citation Context ...ting [Lifschitz et al. 1999], where the closeness between nesting in logic programs and the logic MKNF suggests that an embedding is straightforward, and aggregates [Faber et al. 2004; Ferraris 2005; =-=Pelov et al. 2007-=-; Son and Pontelli 2007]. ACM Transactions on Computational Logic, Vol. V, No. N, Month 20YY.40 · Jos de Bruijn et al. Furthermore, in the present work, we considered embeddings of logic programs int... |

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41 | Uniform semantic treatment of default and autoepistemic logics
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Citation Context ...antics for logic programs with negation that has also been considered for combination of rules and ontologies [Knorr et al. 2008; Drabent et al. 2007]. Three-valued extensions of autoepistemic logic [=-=Denecker et al. 2003-=-; Bonatti 1995; Przymusinski 1991b] may be used as a starting point. Lastly, the initial motivation for our work has been the application to Semantic Web languages. Combinations of positive RIF rules ... |

39 | Can owl and logic programming live together happily ever after - Motik, Horrocks, et al. |

35 | G.: Extended logic programs as autoepistemic theories
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Citation Context ...ates are to be minimized. Embedding a classical theory in AEL is trivial, and several embeddings of logic programs in AEL have been described [Gelfond and Lifschitz 1988; Marek and Truszczynski 1993; =-=Lifschitz and Schwarz 1993-=-; Chen 1993; Przymusinski 1991a]. However, they all have been developed for the propositional case only, whereas we need to deal with non-ground theories and programs. This requires us to consider fir... |

34 | Three-Valued Non-Monotonic Formalisms And Semantics of Logic Programs. Artificial intelligence, 49, 309–343. (Extended abstract appeared in: Three-valued non-monotonic formalisms and logic programming
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Citation Context ...lassical theory in AEL is trivial, and several embeddings of logic programs in AEL have been described [Gelfond and Lifschitz 1988; Marek and Truszczynski 1993; Lifschitz and Schwarz 1993; Chen 1993; =-=Przymusinski 1991-=-a]. However, they all have been developed for the propositional case only, whereas we need to deal with non-ground theories and programs. This requires us to consider first-order autoepistemic logic (... |

29 | On the decidability and complexity of autoepistemic reasoning - Niemelä - 1992 |

27 |
Quantification in Autoepistemic Logic
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(Show Context)
Citation Context ...er, they all have been developed for the propositional case only, whereas we need to deal with non-ground theories and programs. This requires us to consider first-order autoepistemic logic (FO-AEL) [=-=Konolige 1991-=-; Kaminski and Rey 2002; Levesque and Lakemeyer 2000], and non-ground versions of these embeddings. We consider the semantics for FO-AEL as defined by Konolige [1991], because it faithfully extends fi... |

27 |
A constructive semantic characterization of aggregates in answer set programming. Theory and Practice of Logic Programming
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Citation Context ...al. 1999], where the closeness between nesting in logic programs and the logic MKNF suggests that an embedding is straightforward, and aggregates [Faber et al. 2004; Ferraris 2005; Pelov et al. 2007; =-=Son and Pontelli 2007-=-]. ACM Transactions on Computational Logic, Vol. V, No. N, Month 20YY.40 · Jos de Bruijn et al. Furthermore, in the present work, we considered embeddings of logic programs interpreted under the stab... |

26 | Autoepistemic modal logics
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Citation Context ...mbinations. Autoepistemic logic (AEL) [Moore 1985], which extends classical logic with a single nonmonotonic modal belief operator, being essentially the nonmonotonic variant of the modal logic kd45 [=-=Shvarts 1990-=-; Marek and Truszczyński 1993], is an attractive candidate for serving as a uniform host formalism for combinations. Compared to other well-known nonmonotonic formalisms, like Reiter’s default logic [... |

23 | Guarded open answer set programming - Heymans, Nieuwenborgh, et al. - 2005 |

23 |
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Citation Context ...ively, i.e., Γo = Γ ∩ L, Γog = Γ ∩ Lg, and Γoga = Γ ∩ Lga. ACM Transactions on Computational Logic, Vol. V, No. N, Month 20YY.10 · Jos de Bruijn et al. Every stable expansion T of Φ is a stable set [=-=Stalnaker 1993-=-], which means that it satisfies the following conditions: (a) T is closed under first-order entailment, (b) if φ ∈ T then Lφ ∈ T , and (c) if φ /∈ T then ¬Lφ ∈ T . Furthermore, if T is consistent, th... |

19 | On representational issues about combinations of classical theories with nonmonotonic rules - Bruijn, Eiter, et al. - 2006 |

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Citation Context ...ine, in contrast, extensions of the semantics of logic programs or ontologies to incorporate (parts of) the other formalism. One such extension of logic programming semantics is that of open domains [=-=Gelfond and Przymusinska 1993-=-]. Such extended semantics can be used to accommodate incomplete knowledge, an important aspect of ontology languages. Van Belleghem et al. [1997] define open logic programs, which are combinations of... |