  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.
