| 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: | |
| $$ | |
| \begin{array}{lcl} | |
| \textit{circle} & \dot{\sqsubseteq} & \textit{figure} \\ | |
| \textit{figure touching } a\textit{-figure} & \doteq & \textit{figure} \sqcap \exists \textit{EC}. \textit{figure} \\ | |
| \textit{special figure} & \doteq & \textit{figure} \sqcap \\ | |
| & & \forall \textit{PO}. \neg \textit{figure} \sqcap \\ | |
| & & \forall \textit{NTPPI}. \neg \textit{figure} \sqcap \\ | |
| & & \forall \textit{TPPI}. \neg \textit{circle} \sqcap \\ | |
| & & \exists \textit{TPPI}. (\textit{figure} \sqcap \exists \textit{EC}. \textit{circle}) | |
| \end{array} | |
| $$ |