Abstract
The computational complexity of reasoning on pure concept expressions has been characterized completely for all relevant description logics. On the contrary, reasoning in the presence of schema axioms is not so well understood and far from being settled completely. An important class of schemata is that of primitive schemata (in which the schema axioms express only necessary conditions) possibly containing cycles. In this paper we provide, for a relevant class of description logics, a complete characterization of computational complexity of reasoning in these types of schemata, both in the presence and in the absence of cycles. The results are obtained by devising reasoning procedures, establishing direct reductions to show lower bounds, and introducing a general technique by which the constructor for existential quantification can be removed without influencing the result of reasoning.