Rewriting ontology mediated queries (OMQs) into traditional query languages like FO-queries and Datalog is central for understanding their relative expressiveness, and plays a crucial role in the ongoing efforts to develop OMQ answering tools by reusing existing database technologies. However, the vast majority of works focus on Horn ontologies, and for OMQs where the ontologies are written in extensions of ALC, only a couple of exponential rewritings into disjunctive Datalog are known. In this

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... paper we propose a translation of instance queries mediated by ontologies in the expressive DL ALCHI, into polynomial-sized disjunctive Datalog programs. The translation is based on a simple game-like algorithm, and can be extended to accommodate nominals. We can also rewrite OMQs with closed-predicates into Datalog programs with (limited) negation. Closed predicates are useful for combining complete and incomplete information, but make OMQs non-monotonic and thus not rewritable into positive disjunctive Datalog.