  Concrete domains, especially those that allow to compare features with
numeric values, have long been recognized as a very desirable extension of
description logics (DLs), and significant efforts have been invested into
adding them to usual DLs while keeping the complexity of reasoning in check.
For expressive DLs and in the presence of general TBoxes, for standard
reasoning tasks like consistency, the most general decidability results are for
the so-called $\omega$-admissible domains, which are required to be dense.
Supporting non-dense domains for features that range over integers or natural
numbers remained largely open, despite often being singled out as a highly
desirable extension. The decidability of some extensions of $\mathcal{ALC}$
with non-dense domains has been shown, but existing results rely on powerful
machinery that does not allow to infer any elementary bounds on the complexity
of the problem. In this paper, we study an extension of $\mathcal{ALC}$ with a
rich integer domain that allows for comparisons (between features, and between
features and constants coded in unary), and prove that consistency can be
solved using automata-theoretic techniques in single exponential time, and thus
has no higher worst-case complexity than standard $\mathcal{ALC}$. Our upper
bounds apply to some extensions of DLs with concrete domains known from the
literature, support general TBoxes, and allow for comparing values along paths
of ordinary (not necessarily functional) roles.
