An ExpTime Upper Bound for $\mathcal{ALC}$ with Integers (Extended Version)
This addresses a long-standing open issue in knowledge representation for AI, enabling more expressive reasoning with integer features without increasing worst-case complexity, though it is incremental relative to existing dense domain results.
The paper tackles the problem of extending description logics with non-dense integer domains while maintaining computational tractability, proving that consistency for an extension of ALC with rich integer comparisons can be solved in single exponential time, matching the complexity of standard ALC.
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 $ω$-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.