Abstract
Two main families of ontology languages are considered in the context of data access, namely Horn description logics and existential rules. In this paper, we review the semantic relationships between these families in the light of the ontology-mediated query answering problem. To this end, we rely on the standard translation of description logics in first-order logic and on the notion of semantic emulation. We focus on description logics and classes of existential rules for which the conjunctive query answering problem has polynomial data complexity.
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Notes
If we make the simplifying assumption that I interprets constants by themselves, a match of q in I can be seen as a homomorphism from q to the instance naturally associated with I.
The framework in [10] considers more general equality rules, where \(x_1\) and \(x_2\) may also be constants.
Note that the core chase is not defined when it does not terminate [25].
First, computing the core of an extended instance is difficult. The associated decision problem is both NP-hard and coNP-hard, precisely DP-complete [30]. Second, the extended instance built by the core chase does not grow monotonically, which in practice does not allow to update it incrementally.
The equivalence between the rewritability into a union of CQs and first-order rewritability follows from the (Finite) Homomorphism Preservation Theorem [52].
The decidability of query answering in the bts class follows from a theorem by Courcelle [27], which states that satisfiability is decidable for any class of first-order formulas enjoying the bounded-treewidth model property (i.e., satisfiability implies the existence of a model with a bounded-treewidth).
Constants may occur in the initial instance but they may also be brought by the rules, since rule heads may contain constants.
For ba-fr1, 2ExpTime-hardness is still open in the unbounded arity case, while ExpTime-completeness holds in the bounded arity case [53].
To simplify the discussion, we chose not to include nominals nor role composition (except for its restricted use in the transitivity axiom which could be written \(r \circ r \sqsubseteq r\)).
Surpringly, ba-fr1 is not in a lower complexity class than ba-fg for data and bounded-arity combined complexity.
Actually, this result is established for a larger first-order logic fragment, itself included in the so-called guarded negation fragment.
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Acknowledgements
Many thanks to the anonymous reviewers for their helpful comments and suggestions. This work was partially funded by the ANR project CQFD (ANR-18-CE23-0003).
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Mugnier, ML. Data Access With Horn Ontologies: Where Description Logics Meet Existential Rules. Künstl Intell 34, 475–489 (2020). https://doi.org/10.1007/s13218-020-00678-3
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DOI: https://doi.org/10.1007/s13218-020-00678-3