Fitting Description Logic Ontologies to ABox and Query Examples

We study a fitting problem inspired by ontology-mediated querying: given a collection of positive and negative examples of the form (A, q) with A an ABox and q a query, we seek an ontology O such that A ∪ O entails q for all positive examples (A, q) and A ∪ O does not entail q for all negative examples (A, q). We consider the description logics ALC and ALCI as ontology languages and a range of query languages that includes atomic queries (AQs), conjunctive queries (CQs), and unions thereof (UCQs). For all of the resulting fitting problems, we provide effective characterizations and determine the computational complexity of deciding whether a fitting ontology exists. This problem turns out to be coNP-complete for AQs and full CQs and 2ExpTime-complete for CQs and UCQs. These results hold for both ALC and ALCI.