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The description logic with atomic concepts, top and bottom, Boolean concept constructors, and existential and universal restrictions on description logic roles.

A unary description interpreted as a subset of the domain. Atomic names can be combined using the constructors of the chosen description logic.

A binary relation on individuals used by description logic restrictions such as $\exists R.C$ and $\forall R.C$.

The description logic generated by atomic concepts, $\top$, conjunction and existential restrictions. Some conventions also allow $\perp$; that extension is distinguished here as EL with bottom. Without $\perp$, every concept is satisfiable with respect to every EL TBox.

The standard semantics of ALC and EL do not impose the unique name assumption. Therefore the two individual names may denote the same element. A minimum-size model of either normalized TBox from part (d), together with the ABox, isThe $R$ self-loop supplies the ABox fact and makes both $\exists R.A$ and $\exists R.X$ true at $d$. The $P$ self-loop supplies exactly the existential restriction defining $C$. Disjointness holds because $B$ is empty. Hence every normalized general concept inclusion is satisfied. A domain must be nonempty, so $\boxed{|{\Delta }^{\mathcal{I}}|=1}$ is minimal.

If a separate unique name assumption were imposed, the minimum would instead be two. One can take $a\ne b$, $A=\{a\}$, $C=\{b\}$, $R=\{(a,b)\}$, $P=\{(b,b)\}$ and $B=X=\varnothing$. There is then no $R$-edge ending at an $A$ element, so the normalized requirements involving $X$ hold vacuously. That additional convention is not part of standard description logic semantics.

A rule-oriented description logic fragment designed for translation to function-free Horn rules, allowing constraints but excluding existential witness generation in rule conclusions. In an equivalence, each direction must meet the relevant restrictions. ^[ref]