Horn-Rewritability vs PTime Query Evaluation in Ontology-Mediated Querying

In ontology-mediated querying with an expressive description logic L, two desirable properties of a TBox T are (1) being able to replace T with a TBox formulated in the Horn-fragment of L without affecting the answers to conjunctive queries, and (2) that every conjunctive query can be evaluated in PTime w.r.t. T. We investigate in which cases (1) and (2) are equivalent, finding that the answer depends on whether the unique name assumption (UNA) is made, on the description logic under consideration, and on the nesting depth of quantifiers in the TBox. We also clarify the relationship between query evaluation with and without UNA and consider natural variations of property (1).