Containment for Rule-Based Ontology-Mediated Queries

Many efforts have been dedicated to identify restrictions on ontologies expressed as tuple-generating dependencies (tgds), a.k.a. existential rules, that lead to the decidability for the problem of answering ontology-mediated queries (OMQs). This has given rise to three families of formalisms: guarded, non-recursive, and sticky sets of tgds. In this work, we study the containment problem for OMQs expressed in such formalisms, which is a key ingredient for solving static analysis tasks associated with them. Our main contribution is the development of specially tailored techniques for OMQ containment under the classes of tgds stated above. This enables us to obtain sharp complexity bounds for the problems at hand. We start with OMQs based on linear (a core fragment of guarded tgds), non-recursive and sticky sets of tgds, which share the property of UCQ rewritability. By exploiting this property, we obtain a small witness property for non-containment that allows us to devise standard “guessand-check” algorithms. Based on this, we prove that OMQ containment for linear tgds is PSpace-complete, but Π2 complete if the arity of the schema is fixed, while for sticky sets of tgds it is coNExpTime-complete, and Π2 -complete in the case of fixed arity. For non-recursive sets of tgds, the problem is in ExpSpace and P-hard even for fixed-arity schemas. Hence, in all the cases the containment problem is harder (under widely accepted complexity assumptions) than query evaluation, with the exception of linear tgds over schemas of unbounded arity, where both problems are complete for PSpace. For OMQs based on guarded tgds, which are not UCQ rewritable in general, we establish a tree witness property, and use it in combination with tree automata techniques to prove that OMQ containment has the same complexity as query evaluation, i.e., 2ExpTimecomplete. Unlike query evaluation, though, OMQ containment for guarded tgds remains 2ExpTime-hard over fixedarity schemas. Lastly, we apply our techniques to pinpoint the complexity of problems associated with two emerging applications of OMQ containment: distribution over components and UCQ rewritability of OMQs.