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 in {o,-,Or,And}. In particular, as far as languages equipped with role composition (o) are concerned, 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 of reasoning.

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 ⊆ {◦, − , ⊔, ⊓}. In particular, they showed in 1999 that, although ALCN(◦) is decidable, the addition of other operators may easily lead to undecidability: in fact, ALCN(◦, ⊓) and ALCN(◦, − , ⊔) 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(◦) 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.

concepts. In particular, our current investigations are focused on two conjectures, partially supported by some preliminary results: 1. Undecidability of ALCN () extended with inverse roles (which is equivalent to ALC N (; )). 2. Decidability of ALCN () extended with quali ed number restrictions (i.e. ALCQ()). To show undecidability of ALC N (; )-concept satis ability and prove (1), we are working on a reduction of the unrestricted domino problem. In this reduction, we are trying to base the reachability of all the grid points that tile the plane on a concept similar to 9 =1 (U R)(U R) u8U:(9Ru8R:(9 n R u8R :9U )), a starting individual s should be instance of. The rst conjunct is used to ensure that s is the unique (U R)-predecessor of any point that can be reached via U R. The second conjunct ensures that there are at least n of such successors, that will play the role of grid points in the construction. Horizontal and vertical successor relations on the grid would