Abstract
Inconsistency in formal ontologies is usually addressed by computing repairs for the dataset. There are several strategies for selecting the repairs used to evaluate queries, with various levels of cautiousness and classes of computational complexity. This paper deals with inconsistent partially ordered lightweight ontologies. It introduces a new method that goes beyond the cautious strategies and that is tractable in the possibilistic setting, where uncertainty concerns only the data pieces. The proposed method, called C\(\pi \)-repair, proceeds as follows. It first interprets the partially ordered dataset as a family of totally ordered datasets. Then, it computes a single data repair for every totally ordered possibilistic ontology induced from the partially ordered possibilistic ontology. Next, it deductively closes each of these repairs in order to increase their productivity, without introducing conflicts or arbitrary data pieces. Finally, it intersects the closed repairs to obtain a single data repair for the initial ontology. The main contribution of this paper is an equivalent characterization that does not enumerate all the total orders, but also does not suffer from the additional computational cost naturally incurred by the deductive closure. We establish the tractability of our method by reformulating the problem using the notions of dominance and support. Intuitively, the valid conclusions are supported against conflicts by consistent inclusion-minimal subsets of the dataset that dominate all the conflicts. We also study the rationality properties of our method in terms of unconditional and conditional query-answering.
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Notes
- 1.
Given two inconsistency-tolerant semantics \(s_1\) and \(s_2\). Then \(s_1\) is more productive than \(s_2\) if any conclusion derived with \(s_2\) can also be derived with \(s_1\).
- 2.
Note the difference between the ICR semantics which deductively closes the repairs of an ABox, and the ICAR semantics which computes the repairs for a closed ABox.
- 3.
A binary relation \(\ge \) over \({\mathcal {A}}\) is a total preorder if it is reflexive and transitive, and for all \(\varphi _j \in {\mathcal {A}}\), for all \(\varphi _k \in {\mathcal {A}}\), either \(\varphi _j \ge \varphi _k\) or \(\varphi _k \ge \varphi _j\).
- 4.
A binary relation \(\unrhd \) over \({\mathcal {A}}\) is a partial preorder if it is reflexive and transitive. Thus somes elements of \({\mathcal {A}}\) may be incomparable according to \(\unrhd \).
- 5.
Consider \(\varphi _j\) and \(\varphi _k\) in \({\mathcal {A}}\). \(\varphi _j \bowtie \varphi _k\) means that neither \(\varphi _j \unrhd \varphi _k\) nor \(\varphi _k \unrhd \varphi _j\) holds.
- 6.
In propositional logic (PL), RM is defined as: if \(\alpha \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \gamma \) and \(\alpha \,{\mid \!\not \sim }_{{\mathcal {K}}}^{}\, \lnot \beta \), then \(\alpha \wedge \beta \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \gamma \), where \(\alpha \), \(\beta \) and \(\gamma \) are PL formulas. Our adaptation consists first in rewriting RM equivalently in a disjunctive way: if \(\alpha \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \gamma \), then \(\alpha \wedge \beta \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \gamma \) or \(\alpha \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \lnot \beta \). Lastly, we replace \(\alpha \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \lnot \beta \) with \(\alpha \wedge \beta \) is inconsistent with the KB.
- 7.
In PL, Comp is defined as: if \(\alpha \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \gamma \), then either \(\alpha \wedge \beta \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \gamma \) or \(\alpha \wedge \beta \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \lnot \gamma \). Here, we simply replace \(\alpha \wedge \beta \,{\mid \!\sim }_{{\mathcal {K}}}^{}\, \lnot \gamma \) with \(\alpha \wedge \beta \wedge \gamma \) is inconsistent with the KB.
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Acknowledgements
This research has received support from the European Union’s Horizon research and innovation programme under the MSCA-SE (Marie Skłodowska-Curie Actions Staff Exchange) grant agreement 101086252; Call: HORIZON-MSCA-2021-SE-01; Project title: STARWARS (STormwAteR and WastewAteR networkS heterogeneous data AI-driven management).
This research has also received support from the French national project ANR (Agence Nationale de la Recherche) EXPIDA (EXplainable and parsimonious Preference models to get the most out of Inconsistent DAtabases), grant number ANR-22-CE23-0017 and from the ANR project Vivah (Vers une intelligence artificielle à visage humain), grant number ANR-20-THIA-0004.
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Laouar, A., Belabbes, S., Benferhat, S. (2023). Tractable Closure-Based Possibilistic Repair for Partially Ordered DL-Lite Ontologies. In: Gaggl, S., Martinez, M.V., Ortiz, M. (eds) Logics in Artificial Intelligence. JELIA 2023. Lecture Notes in Computer Science(), vol 14281. Springer, Cham. https://doi.org/10.1007/978-3-031-43619-2_25
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