We extend the terminological formalism of the well-known description logic ALC from concept inclusions (CIs) to more general constraints expressed in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA). In QFBAPA one can formulate Boolean combinations of inclusion constraints and numerical constraints on the cardinalities of sets. Our new formalism extends, on the one hand, so-called cardinality restrictions on concepts, which have been introduced two decades ago, and on the other hand the recently defined statistical knowledge bases. Though considerably more expressive, our formalism has the same complexity (NExpTime) as cardinality restrictions on concepts. We will also introduce a restricted version of our formalism for which the complexity is ExpTime. This yields the until now unknown exact complexity of the consistency problem for statistical knowledge bases.

In Description Logics (DL) knowledge bases (KBs), information is typically captured by clear-cut concepts. For many practical applications querying the KB by crisp concepts is too restrictive; a user might be willing to lose some precision in the query, in exchange of a larger selection of answers. Similarity measures can offer a controlled way of gradually relaxing a query concept within a user-specified limit.In this paper we formalize the task of instance query answering for DL KBs using concepts relaxed by concept similarity measures (CSMs). We investigate computation algorithms for this task in the DL EL, their complexity and properties for the CSMs employed regarding whether unfoldable or general TBoxes are used. For the case of general TBoxes we define a family of CSMs that take the full TBox information into account, when assessing the similarity of concepts.

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Description Logics (DLs) are a family of knowledge representation formalisms, that provides the theoretical basis for the standard web ontology language OWL. Generalization services like the least common subsumer (lcs) and the most specific concept (msc) are the basis of several ontology design methods, and form the core of similarity measures. For the DL ELOR, which covers most of the OWL 2 EL profile, the lcs and msc need not exist in general, but they always exist if restricted to a given role-depth. We present algorithms that compute these role-depth bounded generalizations. Our method is easy to implement, as it is based on the polynomial-time completion algorithm for ELOR.