We study the complexity of the combination of the Description Logics ALCQ and ALCQI with a terminological formalism based on cardinality restrictions on concepts. These combinations can naturally be embedded into C 2 , the two variable fragment of predicate logic with counting quantiers, which yields decidability in NExpTime. We show that this approach leads to an optimal solution for ALCQI , as ALCQI with cardinality restrictions has the same complexity as C 2 (NExpTime-complete). In contrast, we show that for ALCQ, the problem can be solved in ExpTime. This result is obtained by a reduction of reasoning with cardinality restrictions to reasoning with the (in general weaker) terminological formalism of general axioms for ALCQ extended with nominals . Using the same reduction, we show that, for the extension of ALCQI with nominals, reasoning with general axioms is a NExpTime-complete problem. Finally, we sharpen this result and show that pure concept satisability for A...

We study the complexity of the combination of the Description Logic ALCQI with a terminological formalism based on cardinality restrictions on concepts. This combination can naturally be embedded into C, the two variable fragment of predicate logic with counting quantifiers, which yields a simple decidability result. We prove that with respect to worst-case complexity, this approach leads to an optimal solution, as ALCQI with cardinality restrictions has the same complexity as C² (NExpTime). Moreover, we show how, in the presence of nominals, reasoning with cardinality restrictions can be reduced to reasoning with the weaker terminological formalism of general inclusion axioms. Using this reduction, we obtain two novel complexity results: reasoning for ALCQ with cardinality restrictions is ExpTime-complete, while reasoning for ALCQI with general inclusion axioms and nominals is NExpTime-hard.