Abstract. We propose an approach for extending a tableau-based satisfiability algorithm by an arithmetic component. The result is a hybrid satisfiability algorithm for the Description Logic (DL) ALCQ which extends ALC with qualified number restrictions. The hybrid approach ensures a more informed calculus which, on the one hand, adequately handles the interaction between numerical and logical restrictions of descriptions, and on the other hand, when applied is a very promising framework for average case optimizations. 1

This article presents a hybrid Abox tableau calculus for SHQ which extends the basic description logic ALC with role hierarchies, transitive roles, and qualified number restrictions. The prominent feature of our hybrid calculus is that it reduces reasoning about qualified number restrictions to integer linear programming. The calculus decides SHQ Abox consistency w.r.t. a Tbox containing general axioms. The presented approach ensures a more informed calculus which adequately handles the interaction between numerical and logical restrictions in SHQ concept and individual descriptions. A prototype reasoner for deciding ALCHQ concept satisfiability has been implemented. An empirical evaluation of our hybrid reasoner and its integrated optimization techniques for a set of synthesized benchmarks featuring qualified number restrictions clearly demonstrates the effectiveness of our hybrid calculus.

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.(Science ⊔ Engineering ⊔ Business) ⊓ ( ≥ 120 hasCredit.(Science ⊔ Engineering)) ⊓ ( ≥ 140 hasCredit)⊓( ≤ 32 hasCredit.(Science⊔Business))⊓(≤91 hasCredit.Engineering)

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