Systems for hybrid reasoning with first-order logic (FOL) extensions of description logic (DL) date back at least 20 years and are enjoying a renewed interest in the context of recent FOL extensions of OWL DL for the Semantic Web. However, current systems for reasoning with such languages can only handle subsets of FOL or they do not fully exploit recent advances in both FOL theorem proving and DL inference. In response, we present an ordered theory resolution calculus for hybrid reasoning in unrestricted FOL extensions of the DL SHI. This calculus permits near-seamless integration of highly optimized FOL theorem provers and DL reasoners while minimizing redundant reasoning and maintaining soundness and refutational completeness. Empirical results demonstrate the potential of this approach in comparison to the state-of-the-art FOL theorem provers Vampire, Otter, and SPASS.
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