# Table 17 Syntax and semantics of description logic [18]

Syntax

Description

Semantics

CD

Concepts or Classes

$$C^{{\mathcal {I}}},D^{{\mathcal {I}}}\subseteq \Theta$$

RS

Relations or Properties

$$R^{{\mathcal {I}}},S^{{\mathcal {I}}}\subseteq \Theta \times \Theta$$

$$C\sqsubseteq D$$

Class subsumption

$$C^{{\mathcal {I}}}\subseteq D^{{\mathcal {I}}}$$

$$R\sqsubseteq S$$

Property subsumption

$$R^{{\mathcal {I}}}\subseteq S^{{\mathcal {I}}}$$

$$C\sqcup D$$

Union of classes

$$(C\sqcup D)^{\mathcal {I}}=C^{{\mathcal {I}}}\cup D^{\mathcal {{\mathcal {I}}}}$$

$$C \sqcap D$$

Intersection of classes

$$(C \sqcap D)^{{\mathcal {I}}}=C^{{\mathcal {I}}} \cap D^{{\mathcal {I}}}$$

$$\lnot C$$

Complement

$$(\lnot C)^{\mathcal {I}}=\Theta - C^{{\mathcal {I}}}$$

$$\exists R.C$$

Existential quantification

$${(\exists R.C)}^{\mathcal {I}}=\{a \in \Theta | \exists b, \langle a, b \rangle \in R^{\mathcal {I}} \wedge b \in C^{\mathcal {I}}\}$$

$$\forall R.C$$

Value restriction

$${(\forall R.C)}^{\mathcal {I}}=\{a \in \Theta | \forall b, \langle a, b \rangle \in R^{\mathcal {I}} \rightarrow b \in C^{{\mathcal {I}}}\}$$

$$\top$$

Top concept or Thing

$$\top ^{{\mathcal {I}}}=\Theta$$

$$\bot$$

Bottom concept or Nothing

$$\bot ^{{\mathcal {I}}}=\varnothing$$

1. - The model is $${\mathcal {M}}=\langle \Theta ,.^ {\mathcal {I}} \rangle$$, where
2. $$~~~\Theta$$ is the domain of objects, and
3. $$~~~{\mathcal {I}}$$ is an interpretation function