Franz Baader and Ulrike Sattler
Number restrictions are concept constructors that are available in almost all implemented description logic systems. However, even though there has lately been considerable effort on integrating expressive role constructors into description logics, the roles that may occur in number restrictions are usually of a very restricted type. Until now, only languages with number restrictions on atomic roles and inversion of atomic roles, or with number restrictions on intersection of atomic roles have been investigated in detail. In the present paper, we increase the expressive power of description languages by allowing for more complex roles in number restrictions. As role constructors, we consider composition of roles (which will be present in all our languages), and intersection, union and inversion of roles in different combinations. We will present two decidability results (for the basic language that extends A~U by number restrictions on roles with composition, and for one extension of this language), and three undecidability results for three other extensions of the basic language.