A Complete Classification of the Complexity and Rewritability of Ontology-Mediated Queries based on the Description Logic EL

We provide an ultimately fine-grained analysis of the data complexity and rewritability of ontology-mediated queries (OMQs) based on an EL ontology and a conjunctive query (CQ). Our main results are that every such OMQ is in AC0, NL-complete, or PTime-complete and that containment in NL coincides with rewritability into linear Datalog (whereas containment in AC0 coincides with rewritability into first-order logic). We establish natural characterizations of the three cases in terms of bounded depth and (un)bounded pathwidth, and show that every of the associated meta problems such as deciding wether a given OMQ is rewritable into linear Datalog is ExpTime-complete. We also give a way to construct linear Datalog rewritings when they exist and prove that there is no constant Datalog rewritings.