Figure 5: Illustration of a model of special_figure
Then, the question is: does figure_touching_a_figure subsume special_figure, or equivalently, is figure ⊓ ∀PO.¬figure ⊓ ∀NTPPI.¬figure ⊓ ∀TPPI.¬circle ⊓ ∃TPPI.(figure ⊓ ∃EC.circle) ⊓ ¬(figure ⊓ ∃EC.figure) unsatisfiable w.r.t. a role box $\mathfrak{R}$ corresponding to the RCC8 composition table?
After pushing the negation sign inwards and removing the obviously contradictory disjunct from the resulting disjunction, the concept figure ⊓ ∀PO.¬figure ⊓ ∀NTPPI.¬figure ⊓ ∀TPPI.¬circle ⊓ ∃TPPI.(figure ⊓ ∃EC.circle) ⊓ ∀EC.¬figure must be unsatisfiable then. Please consider Figure 5 which illustrates a “model” of special_figure, with $a \in special_figure^I$, $b \in (figure \sqcap \exists EC circle)^I$, and $c \in circle^I$, with $\langle a, b \rangle \in TPPI^I$, $\langle b, c \rangle \in EC^I$; please note that $TPPI \circ EC \subseteq EC \sqcup PO \sqcup TPPI \sqcup NTPPI \in \mathfrak{R}$. Due to the definition of special_figure, it can be seen that in every model $\langle a, c \rangle \in EC^I$ must hold. But then, due to ∀EC.¬figure, it is obviously the case that figure ⊓ ∀PO.¬figure ⊓ ∀NTPPI.¬figure ⊓ ∀TPPI.¬circle ⊓ ∃TPPI.(figure ⊓ ∃EC.circle) ⊓ ∀EC.¬figure has no models and is therefore unsatisfiable. This shows that special_figure is indeed subsumed by figure_touching_a_figure.
Please note that there are also other description logics suitable for spatial reasoning tasks, namely the language $\mathcal{ALCRP}(D)$ (see [11]). However, unrestricted $\mathcal{ALCRP}(D)$ is undecidable (see [15]), and its decidable fragment suffers from very strong syntax-restrictions, dramatically pruning the space of allowed concept expressions. In fact, the finite model property is ensured in re-