An ExpTime Upper Bound for $\mathcal{ALC}$ with Integers

Concrete domains, especially those that allow to compare features with numeric values, have long been recognized as a very desirable extension of description logics (DLs), and significant efforts have been invested into adding them to usual DLs while keeping the complexity of reasoning in check. For expressive DLs and in the presence of general TBoxes, for standard reasoning tasks like consistency, the most general decidability results are for the so-called $\omega$-admissible domains, which are required to be dense. Supporting non-dense domains for features that range over integers or natural numbers remained largely open, despite often being singled out as a highly desirable extension. The decidability of some extensions of $\mathcal{ALC}$ with non-dense domains has been shown, but existing results rely on powerful machinery that does not allow to infer any elementary bounds on the complexity of the problem. In this paper, we study an extension of $\mathcal{ALC}$ with a rich integer domain that allows for comparisons (between features, and between features and constants coded in unary), and prove that consistency can be solved using automata-theoretic techniques in single exponential time, and thus has no higher worst-case complexity than standard $\mathcal{ALC}$. Our upper bounds apply to some extensions of DLs with concrete domains known from the literature, support general TBoxes, and allow for comparing values along paths of ordinary (not necessarily functional) roles.

Relaxing and Restraining Queries for OBDA

In ontology-based data access (OBDA), ontologies have been successfully employed for querying possibly unstructured and incomplete data. In this paper, we advocate using ontologies not only to formulate queries and compute their answers, but also for modifying queries by relaxing or restraining them, so that they can retrieve either more or less answers over a given dataset. Towards this goal, we first illustrate that some domain knowledge that could be naturally leveraged in OBDA can be expressed using complex role inclusions (CRI). Queries over ontologies with CRI are not first-order (FO) rewritable in general. We propose an extension of DL-Lite with CRI, and show that conjunctive queries over ontologies in this extension are FO rewritable. Our main contribution is a set of rules to relax and restrain conjunctive queries (CQs). Firstly, we define rules that use the ontology to produce CQs that are relaxations/restrictions over any dataset. Secondly, we introduce a set of data-driven rules, that leverage patterns in the current dataset, to obtain more fine-grained relaxations and restrictions.