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Abstract. We present a resolution-based decision procedure for the description logic SHOIQ—the logic underlying the Semantic Web ontology language OWL-DL. Our procedure is goal-oriented, and it naturally extends a similar procedure for SHIQ, which has proven itself in practice. Applying existing techniques for deriving saturation-based decision procedures to SHOIQ is not straightforward due to nominals, number restrictions, and inverse roles—a combination known to cause termination problems. We overcome this difficulty by using the basic superposition calculus, extended with custom simplification rules. 1

We introduce a calculus of stratified resolution, in which special attention is paid to clauses that "define" relations. If such clauses are discovered in the initial set of clauses, they are treated using the rule of definition unfolding, i.e. the rule that replaces defined relations by their definitions. Stratified resolution comes with a powerful notion of redundancy: a clause to which definition unfolding has been applied can be removed from the search space. To prove the completeness of stratified resolution with redundancies, we use a novel combination of Bachmair and Ganzinger's model construction technique and a hierarchical construction of orderings and least fixpoints.

This paper is dedicated to the memory of Harald Ganzinger.

Logic made a major step in 1879, when Gottlob Frege published his ’Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens’, see [26]. In this paper, which marks the start of modern logic, Frege introduced a formal language, (which he called Begriffsschrift), in which mathematical statements

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