A tetrachotomy of ontology-mediated queries with a covering axiom

We are interested in the problem of efficiently determining the data complexity of answering queries mediated by non-Horn description logic ontologies and constructing their optimal rewritings to standard database queries. In general, this problem is known to be extremely complex. In this article, we strip it to the bare bones and focus on conjunctive queries mediated by a simple covering axiom stating that one class is covered by the union of two other classes. We develop a novel technique to prove that, quite surprisingly, deciding first-order rewritability of even such simple ontology-mediated queries is PSpace-hard. The main result of this article is a complete and transparent syntactic AC0/NL/P/coNP tetrachotomy of path queries under the assumption that the covering classes are disjoint. We also obtain a number of syntactic and semantic sufficient conditions (without the path query assumption) for membership in AC0, L, NL, and P.