Circumscription is one of the main approaches for defining non-monotonic description logics (DLs) and the decidability and complexity of traditional reasoning tasks, such as satisfiability of circumscribed DL knowledge bases (KBs) are well understood. For evaluating conjunctive queries (CQs) and unions thereof (UCQs), in contrast, not even decidability had been established. In this paper, we prove decidability of (U)CQ evaluation on circumscribed DL KBs and obtain a rather complete picture of both the combined complexity and the data complexity, for DLs ranging from ALCHIO via EL to various versions of DL-Lite. We also study the much simpler atomic queries (AQs).

Various properties of ontology modules have been studied, such as coverage, self-containment, depletingness, monotonicity, preservation of justifications. These properties are important from a theoretical and practical point of view because they ensure, e.g., that modules have meaningful interfaces, can be used for ontology debugging, or are suitable for computing a meaningful modular structure of an ontology, such as via atomic decomposition (AD). Given one of the many existing module notions, it is not always obvious whether it satisfies a given property, particularly when the module extraction procedure is based on normalization. We investigate several module properties from an abstract point of view with an emphasis on properties relevant for AD. We examine their interrelations, their relation with iterated module extraction, their preservation in normalization-based module notions, and the adjustment of the latter to the requirements of AD. As a case study, we apply our results to modules based on Datalog reasoning (DBMs), which comprise a large family of normalization-based module notions that provide logical guarantees of varying strengths and are thus suitable to a wide range of use cases. This makes DBMs ready to be used for AD and thereby opens AD to new applications.