Optimizing Reasoning with Qualified Number Restrictions in SHQ

1 Introduction Using SHQ one can express number restrictions on role fillers of individuals. A typical question for a DL reasoner would be whether the concept ∀hasCredit. test the satisfiability of such a concept by first satisfying all at-least restrictions, e.g., by creating 260 hasCredit-fillers, of which 120 are instances of (Science Engineering). Eventually, a nondetermin-istic choose-rule assigns to each of these 260 individuals (Science Business) or ¬(Science Business), and Engineering or ¬Engineering. In case an at-most restriction is violated, e.g., a student has more than 91 hasCredit-fillers of Engineering, a nondeterministic merge-rule tries to reduce the number of these individuals by merging a pair of individuals until the upper bound specified in this at-most restriction is satisfied. Searching for a model in such an arithmetically uninformed or blind way is usually very inefficient. Our hybrid calculus is based on a standard tableau for SH [1] modified and extended to deal with qualified number restrictions and works with an inequation solver based on integer linear programming. The tableau rules encode number restrictions into a set of inequations using the so-called atomic decomposition technique [16]. The set of inequations is processed by the inequation solver which finds, if possible, a minimal non-negative integer solution (distribution of role fillers constrained by number restrictions) satisfying the inequations. The tableau rules ensure that such a distribution of role fillers also satisfies the logical restrictions. Since this hybrid algorithm collects all the information about arithmetic expressions before creating any role filler, it will not satisfy any at-least restriction by violating an at-most restriction and there is no need for a mechanism of merging role fillers and its performance is not affected by the values of numbers occurring in number restrictions. Since the solution from the inequation solver satisfies all numerical restrictions imposed by at-least and at-most restrictions, our calculus needs to create only one so-called proxy individual (inspired by [6]) representing a set of role fillers. Considering all these features the proposed hybrid algorithm is well suited to improve average case performance. Furthermore, in [2, Chapter 6] it has been shown that a tableau procedure extended by global caching and an algebraic method similar to the one presented in this paper and in [16, 8] is worst-case optimal for SHIQ. Although the calculus presented in our paper is different from the one in [2] we conjecture that the result presented in [2] can be transferred …