On Decidability of Expressive Description Logics with Composition of Roles in Number Restrictions

Description Logics are knowledge representation formalisms which have been used in a wide range of application domains. Owing to their appealing expressiveness, we consider in this paper extensions of the well-known concept language ALC allowing for number restrictions on complex role expressions. These have been first introduced by Baader and Sattler as ALCN(M) languages, with the adoption of role constructors M subset-of {o,-,And,Or}. In particular, they showed in 1999 that, although ALCN(o) is decidable, the addition of other operators may easily lead to undecidability: in fact, ALCN(o,And) and ALCN(o,-,Or) were proved undecidable. In this work, we further investigate the computational properties of the ALCN family, aiming at narrowing the decidability gap left open by Baader and Sattler's results. In particular, we will show that ALCN(o) extended with inverse roles both in number and in value restrictions becomes undecidable, whereas it can be safely extended with qualified number restrictions without losing decidability.