This chapter is aimed to provide a concise overview of basic notions used in description logics. Description and modal (and also hybrid) logics have a lot in common. In fact, the former can be considered as a notational variant of the latter. For that reason, references to works from the description logic field occur throughout the whole book. A brief explanation of a description logic terminology we refer to in our considerations is therefore in order.

First, let's recall that the basic modal logic is defined over a sig-nature 〈 prop, R 〉, where prop is a denumerable set of propositional variables p1, p2, … and R is a binary relation over the universe of a model. Description logics use the notion of atomic concepts rather than propositional formulas. Let C0 denote a set of all atomic concepts of (an abstract) description logic D. Again, in description logics we talk about roles rather than about accessibility relations. Let R denote a set of all roles of D. We have all at hand to define a syntax of D.

Definition6.1 (Syntax of D). Let C0 be the set of atomic concepts and R be the set of roles in D. We define the set C of all concepts of the language of the description logic D as follows:

C := ⊤ | A | ⌝C | C ⊓ D | C ⊓ D | ∃R.C | ∀R. C | n ≥ R.C | n ≤ R.C,

where A ∈ C0, C, D ∈ C, R ∈ R and n ∈ ℕ.

Before we draw a parallel between concrete modal and description notions, let's define a model for D which is called an interpretation.

Definition 6.2 (Semantics of D).