We study the description logic SQ with number restrictions applicable to transitive roles, extended with either nominals or inverse roles. We show tight 2EXPTIME upper bounds for unrestricted entailment of regular path queries for both extensions and finite entailment of positive existential queries for nominals. For inverses, we establish 2EXPTIME-completeness for unrestricted and finite entailment of instance queries (the latter under restriction to a single, transitive role).

As the ontology component, we consider extensions of the DL ALC, allowing for transitive The use of ontologies to provide background knowledge closure of roles. The study of finite entailment is relevant for enriching answers to queries posed to a database is a for this combination because, unlike for plain CQs, query major research topic in the fields of knowledge representation entailment of CQs with transitive closure is not finitely controllable and reasoning. In this data-centric setting, various even for ALC, and thus finite and unrestricted entailment options for the formalisms used to express ontologies and do not coincide. As a consequence, dedicated algorithmic queries exist, but popular choices are description logics methods and lower bounds need to be developed.

We study the problem of finite ontology mediated query answering (FOMQA), the variant of OMQA where the represented world is assumed to be finite, and thus only finite models of the ontology are considered. We adopt the most typical setting with unions of conjunctive queries and ontologies expressed in description logics (DLs). The study of FOMQA is relevant in settings that are not finitely controllable. This is the case not only for DLs without the finite model property, but also for those allowing transitive role declarations. When transitive roles are allowed, evaluating queries is challenging: FOMQA is undecidable for SHOIF and only known to be decidable for the Horn fragment of ALCIF. We show decidability of FOMQA for three proper fragments of SOIF: SOI, SOF, and SIF. Our approach is to characterise models relevant for deciding finite query entailment. Relying on a certain regularity of these models, we develop automata-based decision procedures with optimal complexity bounds.