Appendix I: Basics of Description Logics

This page defines the basics of description logics by instantiating the components of a logic defined as part of the generalities on logics. In this page, I define a logic that I will call proto-Description-Logic DL as a tuple (SymDL, ForDL, IntDL, ⊨DL) that we define piece by piece in later sections.

Syntax

Symbols

We define the set B = {‘(’, ‘)’, ‘,’, ⊑} of opening, closing parentheses, comma and concept inclusion symbol, an infinite set C disjoint from B whose members are called atomic concepts, an infinite set R disjoint from B and C whose members are called roles, and an infinite set U disjoint from all other sets, whose members are called individuals.

Symbols of DL
The symbols of DL are SymDL = B ⋃ C ⋃ R ⋃ U.

As with propositions in propositional logic, we will write distinct atomic concepts, roles or individuals with distinct sequences of latin characters like so: Vehicle, ownedBy, Europe, etc.

Formulas

formulas of DL

The set ForDL of DL formulas are grouped in three kinds:

  • for all c, d ∈ C, c ⊑ d ∈ ForDL (these formulas are called concept inclusions);
  • for all c ∈ C and a ∈ U, c(a) ∈ ForDL (these formulas are called concept assertions);
  • for all r ∈ R and a, b ∈ U, r(a, b) ∈ ForDL (these formulas are called role assertions).

Note that we use a light shading on formulas in order to clearly distinguish them from other syntactic constructs.

In a DL theory, formulas can be partitioned into two distinguished sets: one that contains all concept inclusions and one that contains all concept assertions and roles assertions.

ABox and TBox
In a DL theory T, the set of concept inclusions is called the TBox (or terminological box) of T, and the set of concept assertions and roles assertions is called the ABox (or assertional box) of T.

Semantics

Interpretations

Interpretations in DL

An interpretation in DL is a pair (Δ, 𝓘) such that:

  • Δ is a non-empty set called the domain of interpretation;

  • 𝓘 is a function defined on the sets of atomic concepts, roles, and individuals such that:

    • for each c ∈ C, 𝓘(c) ⊆ Δ;
    • for each r ∈ R, 𝓘(r) ⊆ Δ × Δ;
    • for each a ∈ U, 𝓘(a) ∈ Δ.

Satisfaction relation

Satisfaction relation in DL

An interpretation (Δ, 𝓘) in DL satisfies:

Families of description logics

The basic description logic DL presented before defines what is common to all description logics. A specific description logic extends DL in different ways: adding constructs to define complex concepts (for instance, a concept defined as the union of two atomatic concepts); or adding other types of formulas (for instance, allowing role inclusion in addition to concept inclusion); adding other types of entities (for instance, allowing the use of numeric values).

Here is a list of typical description logics have been defined due to their interesting computational properties or their practical relevance to some domains or applications: