Answering regular path queries mediated by unrestricted SQ ontologies

Víctor Gutiérrez-Basulto, Yazmín Ibáñez-García, Jean Christoph Jung, Filip Murlak

Abstract: A prime application of description logics is ontology-mediated query answering, with the query language often reaching far beyond instance queries. Here, we investigate this task for positive existential two-way regular path queries and ontologies formulated in the expressive description logic SQu<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">SQ</mi></mrow><mrow is="true"><mi is="true">u</mi></mrow></msub></math>, where SQu<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">SQ</mi></mrow><mrow is="true"><mi is="true">u</mi></mrow></msub></math> denotes the extension of the basic description logic ALC<math><mi mathvariant="script" is="true">ALC</mi></math> with transitive roles (S<math><mi mathvariant="script" is="true">S</mi></math>) and qualified number restrictions (Q<math><mi mathvariant="script" is="true">Q</mi></math>) which can be unrestrictedly applied to both non-transitive and transitive roles (⋅u<math><msub is="true"><mrow is="true"><mo is="true">⋅</mo></mrow><mrow is="true"><mi is="true">u</mi></mrow></msub></math>). Notably, the latter is usually forbidden in expressive description logics. As the main contribution, we show decidability of ontology-mediated query answering in that setting and establish tight complexity bounds, namely 2ExpTime-completeness in combined complexity and coNP-completeness in data complexity. Since the lower bounds are inherited from the fragment ALC<math><mi mathvariant="script" is="true">ALC</mi></math>, we concentrate on providing upper bounds. As main technical tools we establish a tree-like countermodel property and a characterization of when a query is not satisfied in a tree-like interpretation. Together, these results allow us to use an automata-based approach to query answering.