Description Logics with Concrete Domains and Aggregation

We extend different Description Logics by concrete domains (such as integers and reals) and by aggregation functions over these domains (such as min;max; count; sum), which are usually available in database systems. On the one hand, we show that this extension may lead to undecidability of the basic inference problems satisfiability and subsumption. This is true even for a very small Description Logic and very simple aggregation functions, provided that universal value restrictions are present. On the other hand, disallowing universal value restrictions yields decidability of satisfiability, provided that the concrete domain is not too expressive. An example of such a concrete domain is the set of (nonnegative) integers with comparisons (=, , n, ...) and the aggregation functions min;max; count.