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

**Definition 7 (Satisfiability)** A syntactic entity (concept, role box, concept with role box, etc.) is called *satisfiable* iff there is an interpretation which satisfies this entity; i.e. the entity has a model. ■

Then, the satisfiability problem is to decide whether a syntactic entity is satisfi-
able or not.

An important relationship between concepts is the subsumption relationship,
which is a partial ordering on concepts w.r.t. their specificity:

**Definition 8 (Subsumption Relationship)** A concept *D* subsumes a concept *C*, *C* $\subseteq$ *D*, iff $C^I \subseteq D^I$ holds for all interpretations *I*. $\square$

Since $\mathcal{ALC}_{RA\ominus}$ provides a full negation operator, the subsumption problem can
be reduced to the concept satisfiability problem: $C \sqsubseteq D$ iff $C \sqcap \neg D$ is unsatis-
fiable.

It should be noticed that a satisfiability tester for $\mathcal{ALC}_{RA\ominus}$ would also be able
to determine satisfiability resp. subsumption w.r.t. free TBoxes. Each concept
inclusion axiom can be dealt with by a technique called *internalization* (see
[13, 14, 1]). Internalization for $\mathcal{ALC}_{RA\ominus}$ works as follows. Let $(C, \mathfrak{R}, \mathfrak{T})$ be the
concept, role box and free TBox to be tested for satisfiability. Let $R? \in \mathcal{N}_R$ be
some role such that $R? \notin \text{roles}(C) \cup \text{roles}(\mathfrak{R})$. Referring to $R?$ and $(C, \mathfrak{R}, \mathfrak{T})$,
the role box $\mathfrak{R}$ is completed: $\mathfrak{R}' = \mathfrak{R} \cup \{ R \circ S \sqsubseteq R? \mid R, S \in (\{R?\} \cup \text{roles}(\mathfrak{R})),$
$\neg\exists ra \in \mathfrak{R}: \text{pre}(ra) = (R, S) \}$.

Now, $(C, \mathfrak{R}, \mathfrak{T})$ is satisfiable iff $((C \sqcap M_{\mathfrak{T}} \sqcap \forall *. M_{\mathfrak{T}}), \mathfrak{R}')$ is satisfiable, where
$\forall *. M_{\mathfrak{T}}$ is an abbreviation for $\forall (\sqcup_{R \in \text{roles}(\mathfrak{R}')} R). M_{\mathfrak{T}}$. $M_{\mathfrak{T}}$ is the so-called *meta-constraint* corresponding to the TBox $\mathfrak{T}: M_{\mathfrak{T}} =_{def} \sqcap_{C \sqsubseteq D \in \mathfrak{T}} (\neg C \sqcup D)$.

# 3 Relationships to Other Logics

In order to judge the expressive power of $\mathcal{ALC}_{RA\ominus}$ we consider other logics and examine whether they are subsumed¹ by $\mathcal{ALC}_{RA\ominus}$. We briefly sketch the relationships to the most important base description logics offering some form of transitivity. Additionally, (un)decidability results regarding transitivity extensions of the so-called (loosely) guarded fragment of FOPL are briefly discussed

¹We say that a language *A* is “subsumed” by a language *B* (resp. provides the same expressive power) iff the satisfiability problem of *A* can be reduced to the satisfiability problem of *B*.