Formal ontologies are of significant importance in artificial intelligence, playing a central role in the Semantic Web, ontology-based information integration, or peer-to-peer data management. In such scenarios, an especially prominent role is played by description logics (DLs) – a robust family of logical formalisms used to describe ontologies and serving as the logical underpinning of contemporary standardised ontology languages. To put knowledge bases to full use as core part of intelligent information systems, much attention is being devoted to the area of ontology-based data-access, with conjunctive queries and their generalisations such as positive conjunctive two-way regular path queries being employed as a fundamental querying formalism. The most expressive exemplars of description logics feature advanced constructors for roles and path expressions. Among the most powerful knowledge representation formalisms on the verge of decidability, are the DLs from the Z family. For its most expressive proponent, ZOIQ (a.k.a. ALCHbSelf reg OIQ), featuring nominals (O), role inverses (I), and number restrictions (Q), querying is undecidable and even decidability of knowledge-base satisfiability is open, owing to the intricate interplay of the three mentioned features. Restricting the interaction of O, I, and Q however (or excluding one of the features altogether) leads to beneficial model-theoretic properties, which give rise to upper bounds of ExpTime for knowledge-base satisfiability and 2ExpTime for querying.
Aiming for better understanding of the “expressive power versus computational complexity” trade-off for the Z family of DLs, we provide a more fine-grained complexity analysis for the query entailment problem over ontologies. In the thesis we focus on tame fragments of ZOIQ, namely the fragments in which either one the three features from {I, O, Q} is dropped or the class of models is restricted to the so-called quasi-forests. We employ the query languages ranging from (unions of) conjunctive queries ((U)CQs) to positive two-way regular path queries (P2RPQs). We mostly follow the classical semantics of entailment, but we also provide several results in the “finite-model” scenario. The most important results of the thesis are summarised below.
1. We provide a complete classification of the complexity of the query entailment problem (for various query languages discussed above) for tamed fragments of ZOIQ under the classical semantics. This involves several new ingredients such as: (i) a uniform exponential-time algorithm based on Lutz’s spoiler technique for the entailment of unions of conjunctive queries for ALCHbreg, (ii) new lower bounds for (rooted and unrooted) conjunctive query entailment over ALCSelf ontologies, and (iii) a novel reduction from the entailment of P2RPQs to the satisfiability problem for tamed ZOIQ, yielding a uniform 2ExpTime upper bound for all the considered logics. As a preliminary step towards lifting the above results to the realm of data complexity, we establish that the satisfiability of tamed ZOIQ is NP-complete. 2. Under the finite model semantics, we focus on UCQs only. With the proviso that the finite satisfiability problem for ZIQ is ExpTime-complete, we also provide a complete picture of the complexity of the query entailment problems. The key insight is that ZOI and ZOQ are finitely controllable.
3. We conclude the thesis by investigating the decidability border of further extensions of the Z family of DLs. Our goal is to understand whether the class of regular languages present in path expressions in Z is maximal for guaranteeing decidability of the underlying logic. We provide a series of undecidability results involving simple, non-regular languages (a subclass of visibly pushdown languages).
Our proofs rely on well-established model- and graph-theoretic definitions. What is more, most of them generalise (in a uniform way) and solidify multiple results known by the DL community. Our proofs are also easily adjustable to freshly defined logics, without the need to reproduce nearly-identical proofs.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:92537 |
| Date | 19 July 2024 |
| Creators | Bednarczyk, Bartosz |
| Contributors | Rudolph, Sebastian, Kieroński, Emanuel, Ortiz, Magdalena, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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