Dichotomies in Ontology-Mediated Querying with the Guarded Fragment

Abstract

We study ontology-mediated querying in the case where ontologies are formulated in the guarded fragment of first-order logic (GF) or extensions thereof with counting and where the actual queries are (unions of) conjunctive queries. Our aim is to classify the data complexity and Datalog rewritability of query evaluation depending on the ontology O , where query evaluation w.r.t. O is in PT ime (resp. Datalog rewritable) if all queries can be evaluated in PT ime w.r.t. O (resp. rewritten into Datalog under O ), and co NP-hard if at least one query is co NP-hard w.r.t. O . We identify several fragments of GF that enjoy a dichotomy between Datalog-rewritability (which implies PT ime ) and co NP-hardness as well as several other fragments that enjoy a dichotomy between PT ime and co NP-hardness, but for which PT ime does not imply Datalog-rewritability. For the latter, we establish and exploit a connection to constraint satisfaction problems. We also identify fragments for which there is no dichotomy between PT ime and co NP. To prove this, we establish a non-trivial variation of Ladner’s theorem on the existence of NP-intermediate problems. Finally, we study the decidability of whether a given ontology enjoys PT ime query evaluation, presenting both positive and negative results, depending on the fragment.