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Definition 4 (Generalized Concept Inclusion Axiom, TBox) If C and D are $\mathcal{ALC}_{RA}\ominus$ concepts, then the expression $C \dot{\sqsubseteq} D$ is called a generalized concept inclusion axiom, or GCI for short. A finite set of such GCIs is called a free TBox, $\mathfrak{T}$. We use $C \doteq D \in \mathfrak{T}$ as a shorthand for ${C \dot{\sqsubseteq} D, D \dot{\sqsubseteq} C} \subseteq \mathfrak{T}$. $\square$

The semantics of an $\mathcal{ALC}_{RA}\ominus$ concept is specified by giving a Tarski-style interpretation $\mathcal{I}$ that has to satisfy the following conditions:

Definition 5 (Interpretation) An interpretation $\mathcal{I} =_{def} (\Delta^{\mathcal{I}}, \cdot^{\mathcal{I}})$ consists of a non-empty set $\Delta^{\mathcal{I}}$, called the domain of $\mathcal{I}$, and an interpretation function $\cdot^{\mathcal{I}}$ that maps every concept name to a subset of $\Delta^{\mathcal{I}}$, and every role name to a subset of $\Delta^{\mathcal{I}} \times \Delta^{\mathcal{I}}$.

The interpretation function $\cdot^{\mathcal{I}}$ can then be extended to arbitrary concepts $C$ by using the following definitions (we write $X^{\mathcal{I}}$ instead of $\cdot^{\mathcal{I}}(X)$):

$\begin{align*} (\neg C)^{\mathcal{I}} &=_{def} \Delta^{\mathcal{I}} \setminus C^{\mathcal{I}} \\ (C \sqcap D)^{\mathcal{I}} &=_{def} C^{\mathcal{I}} \cap D^{\mathcal{I}} \\ (C \sqcup D)^{\mathcal{I}} &=_{def} C^{\mathcal{I}} \cup D^{\mathcal{I}} \\ (\exists R.C)^{\mathcal{I}} &=_{def} \{i \in \Delta^{\mathcal{I}} | \exists j \in C^{\mathcal{I}} : <i,j> \in R^{\mathcal{I}}\} \\ (\forall R.C)^{\mathcal{I}} &=_{def} \{i \in \Delta^{\mathcal{I}} | \forall j : <i,j> \in R^{\mathcal{I}} \Rightarrow j \in C^{\mathcal{I}}\} \quad \square \end{align*}$

It is therefore sufficient to provide the interpretations for the concept names and the roles, since the interpretation of every concept is uniquely determined then by using the definitions.

In the following we specify under which conditions a given interpretation is a model of a syntactic entity (we also say an interpretation satisfies a syntactic entity):

Definition 6 (Model Relationship) An interpretation $\mathcal{I}$ satisfies / is a model of a concept $C$, written $\mathcal{I} \models C$, iff $C^{\mathcal{I}} \neq \emptyset$.

An interpretation $\mathcal{I}$ satisfies / is a model of a role axiom $S \circ T \subseteq R_1 \sqcup \cdots \sqcup R_n$, written $\mathcal{I} \models S \circ T \subseteq R_1 \sqcup \cdots \sqcup R_n$, iff $S^{\mathcal{I}} \circ T^{\mathcal{I}} \subseteq R_1^{\mathcal{I}} \cup \cdots \cup R_n^{\mathcal{I}}$.

An interpretation $\mathcal{I}$ satisfies / is a model of a role box $\mathfrak{R}$, written $\mathcal{I} \models \mathfrak{R}$, iff for all role axioms $ra \in \mathfrak{R}: \mathcal{I} \models ra$.

An interpretation $\mathcal{I}$ satisfies / is a model of a GCI $C \dot{\sqsubseteq} D$, written $\mathcal{I} \models C \dot{\sqsubseteq} D$, iff $C^{\mathcal{I}} \subseteq D^{\mathcal{I}}$.

An interpretation $\mathcal{I}$ satisfies / is a model of a TBox $\mathfrak{T}$, written $\mathcal{I} \models \mathfrak{T}$, iff for all GCIs $g \in \mathfrak{T}: \mathcal{I} \models g$.

An interpretation $\mathcal{I}$ satisfies / is a model of $(C, \mathfrak{R})$, written $\mathcal{I} \models (C, \mathfrak{R})$, iff $\mathcal{I} \models C$ and $\mathcal{I} \models \mathfrak{R}$.