Foundations of Description Logics

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Abstract

This chapter accompanies the foundational lecture on Description Logics (DLs) at the 7th Reasoning Web Summer School in Galway, Ireland, 2011. It introduces basic notions and facts about this family of logics which has significantly gained in importance over the recent years as these logics constitute the formal basis for today’s most expressive ontology languages, the OWL (Web Ontology Language) family.

We start out from some general remarks and examples demonstrating the modeling capabilities of description logics as well as their relation to first-order predicate logic. Then we begin our formal treatment by introducing the syntax of DL knowledge bases which comes in three parts: RBox, TBox and ABox. Thereafter, we provide the corresponding standard model-theoretic semantics and give a glimpse of the alternative way of defining the semantics via an embedding into first-order logic with equality.

We continue with an overview of the naming conventions for DLs before we delve into considerations about different notions of semantic alikeness (concept and knowledge base equivalence as well as emulation). These are crucial for investigating the expressivity of DLs and performing normalization. We move on by reviewing knowledge representation capabilities brought about by different DL features and their combinations as well as some model-theoretic properties associated thereto.

Subsequently, we consider typical reasoning tasks occurring in the context of DL knowledge bases. We show how some of these tasks can be reduced to each other, and have a look at different algorithmic approaches to realize automated reasoning in DLs.

Finally, we establish connections between DLs and OWL. We show how DL knowledge bases can be expressed in OWL and, conversely, how OWL modeling features can be translated into DLs.

In our considerations, we focus on the description logic \(\mathcal{SROIQ}\) which underlies the most recent and most expressive yet decidable version of OWL called OWL 2 DL. We concentrate on the logical aspects and omit data types as well as extralogical features from our treatise. Examples and exercises are provided throughout the chapter.