Optimizing the Nominal Introduction Rule in (Hyper)Tableau Calculi

A tableau calculus for SHIQ has been known for some time, and forms the basis for several highly successful implementations [1, 8, 9]. Extending this calculus to SHOIQ, however, was notoriously difficult due to an interaction between nominals, inverse roles, and number restrictions which results in (partial) loss of the forest-model property for models of SHOIQ knowledge bases. In this paper, we analyze the problems that arise from this combination of features and present an overview of two solutions. The first solution, described in [3], extends the SHIQ calculus with an NNrule which extends the non-forest-like portion of a model with new individuals. Applications of this rule can be highly nondeterministic, however, and can substantially increase the size of the generated models, which leads to inefficiency. We present an alternative solution based on a new NI-rule. Our approach does not introduce new individuals; instead, existing individuals are incorporated into the non-forest-like portion of the model. We implemented our solution in HermiT. As we discuss in [7], this approach seems to work well in practice.