An Update on Query Answering with Restricted Forms of Negation

One of the most prominent applications of description logic ontologies is their use for accessing data. In this setting, ontologies provide an abstract conceptual layer of the data schema, and queries over the ontology are then used to access the data. In this paper we focus on extensions of conjunctive queries (CQs) and unions of conjunctive queries (UCQs) with restricted forms of negations such as inequality and safe negation. In particular, we consider ontologies based on members of the DL-Lite family. We show that by extending UCQs with any form of negated atoms, the problem of query answering becomes undecidable even when considering ontologies expressed in the core fragment of DL-Lite. On the other hand, we show that answering CQs with inequalities is decidable for ontologies expressed in DL-$Lite^{\mathcal{H}}_{core}$. To this end, we provide an algorithm matching the known coNP lower bound on data complexity. Furthermore, we identify a setting in which conjunctive query answering with inequalities is tractable. We regain tractability by means of syntactic restrictions on the queries, but keeping the expressiveness of the ontology.