**3.1.2 The description logics formalization**

220 Semantics – Advances in Theories and Mathematical Models

⊤ ��� � ⊤ ⊤(�) Universal concept ⊥ ⊥ ⊥ (�) Bottom concept ⊓ C⊓� C(x) ⊓ D(x) Intersection ⊔ �⊔� C(x) ⊔ D(x) Union ¬ ¬� ¬�(�) Negation <sup>∃</sup> ∃�� � ∃�(�� �)��(�) Existential

∀ ∀�� � ∀�� �(�� �) � �(�) Value Restriction

We introduce in this section the terminological axioms, which make statements about how concepts or roles are related to each other. Then we single out definitions as specific axioms and identify terminologies as sets of definitions by which we can introduce atomic concepts as abbreviations or names for complex concepts. In the most general case, terminological axioms have the form �� � ��� �� � ������ � �� � � � where C, D are concepts (and R, S are roles). Axioms of the first kind are called inclusions, while axioms of the second kind are called equalities. An equality whose left-hand side is an atomic concept. It´s used to introduce symbolic names for complex descriptions e,g. ���������� � ������� ⊓ ∃�������� ��������. It could be clearly seen within Fig 2 that these concept descriptions are built with the concept constructors. The first four constructors are not dependent on the roles whereas the last two utilizes the roles in the constructors. This dependency is called role restrictions. Formally, a role restriction is an unnamed class containing all individuals that satisfy the restriction. DLs expressed through *ALC* provide two such restrictions in

It´s again classified as the **existential quantifier** (at least one, or some) and **universal** 

The existential quantifier links a restriction concept to a concept description or a data range. This restriction describes the unnamed concept for which there should be at least one instance of the concept description or value of the data value. Simplifying, the property restriction *P* relates to a concept of individuals *x* having at least one y which is either an instance of concept description or a value of data range so that *P(x,y)* is an instance of *P*.

From the other side, the **universal quantifier** () (*every)* constraint links a restriction concept to a concept description or a data range. This restriction describes the unnamed concept for which there should all instances of the concept description or value of the data value. Simplifying, the property restriction *P* relates to a concept of individuals *x* having all y which is either an instance of the concept description or a value of data range so that *P(x,y)* 

It links a restriction concept directly to a value which could be either an individual or data

Here C and D are concept description and R is role Fig. 2. The syntax and semantics based on *ALC.*

Quantifier restriction and value restrictions.

**The Quantifier restriction** 

is concidered as an instance of *P*.

**The Value restriction** 

value.

**quantifiers** (every).

Quantification

**Notation Syntax Semantics Read-as** 

Description logics (DLs) are a family of logics which represents the structured knowledge. The Description Logic languages are knowledge representation languages that can be used to represent the knowledge of an application domain in a structured and formally wellunderstood way (McGuinness, et al., 2003), (Calvanese, et al., 2005). Description logics contain the formal, logic-based semantics, which present the major reason for its choice for Semantic Web languages over its predecessors. The reasoning capabilities within the DLs add a new dimension. Having these capabilities as central theme, inferring implicitly represented knowledge becomes possible. The movement of Description Logic into its applicability can be viewed in terms of its progression in Web environment (Noy, et al., 2001). Web languages such as XML or RDF(S) could benefit from the approach DL takes to formalize the structured knowledge representation (Lassila, 2007). This has laid background behind the emergence of Description Logic languages in Web. Actually, an agreement to encode these operators using an alphabetic letter to denote expressivity of DLs has seen the light. These letters in combinations are used to define the capabilities of DLs in terms of their performances. This implies to the DL languages as well. As could be seen in Fig 3, *ALC* has been extended to transitive role and given abbreviation S in the convention. Where S is used in every DL systems and languages as it plays significant role in shaping the behavioural nature of every DL systems.


Fig. 3. Naming convention of Description Logic.
