Decidability of Expressive Description Logics with Role Compositions (Extended Abstract)

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 woul