Figure 4: Illustration of $\forall a, b, c : EC(a, b) \land EC(b, c) \Rightarrow (DC(a, c) \lor EC(a, c) \lor PO(a, c) \lor TPP(a, c) \lor TPPI(a, c))$
means of a so called composition table that lists, given the “column” relationship $R(a, b)$ and the “row” relationship $S(b, c)$, all possible relationships $T_1(a, c)$, $T_2(a, c), ..., T_n(a, c)$ that may hold between $a$ and $c$. For example, in the case of RCC8, the composition table contains the entry ${DC, EC, PO, TPP, TPPI}$, given the relationship EC for the row as well as for the column – please consider Figure 4. This corresponds to the FOPL axiom $\forall a, b, c : EC(a, b) \land EC(b, c) \Rightarrow (DC(a, c) \lor EC(a, c) \lor PO(a, c) \lor TPP(a, c) \lor TPPI(a, c))$, which is equivalent to the role axiom $EC \circ EC \sqsubseteq DC \sqcup EC \sqcup PO \sqcup TPP \sqcup TPPI$.
Usually, also the disjointness of the base relations must be captured. As already noted, $\mathcal{ALC}{RA\ominus}$ lacks this expressiveness (and it cannot be simulated by means of other constructs easily, see below), but $\mathcal{ALC}{RA}$ does not. For an adequate modeling of spatial relationships, also inverse roles must be taken into account. For example, the RCC8 relationship TPPI is the inverse of TPP, and NTPPI is the inverse of NTPP. Of course, $TPP^I = (TPPI^I)^{-1}$ and $NTPP^I = (NTPPI^I)^{-1}$ should be ensured. However, both $\mathcal{ALC}{RA}$ and $\mathcal{ALC}{RA\ominus}$ lack inverse roles, since undecidability would follow immediately then by previously known undecidability results. Since we use $\mathcal{ALC}_{RA\ominus}$ in the following example, we can neither rely on $TPP^I = (TPPI^I)^{-1}$ nor on the disjointness of roles.
The possibility to approximate composition tables, which are very widely used in the field of relation algebra-based knowledge representation and reasoning, is the distinguishing feature of $\mathcal{ALC}{RA\ominus}$ and $\mathcal{ALC}{RA}$. Usually, the $\mathcal{ALC}{RA}$ approximation will be better, since the disjointness of the base relations is also enforced. Nevertheless, as the example demonstrates, we can still solve some interesting spatial reasoning task using $\mathcal{ALC}{RA\ominus}$. Consider the following TBox: