Figure 3: RCC8 Qualitative Spatial Relationships: EQ = Equal, DC = Disconnected, EC = Externally Connected, PO = Partial Overlap, TPP = Tangential Proper Part, NTPP = Non-Tangential Proper Part. Read the relations as TPP(A, B), NTPP(A, B) etc. TPP and NTPP have corresponding inverse relationships: TPPI and NTPPI, e.g. TPPI(B, A), NTPPI(B, A).
• Even $GF^2$ with transitive relations is undecidable (see [7]).
• Monadic $GF^2$ with binary transitive, symmetric and/or reflexive relations is decidable (see [7]).
None of these results is applicable in the case of $\mathcal{ALC}{RA\ominus}$. The most important result concerning $\mathcal{ALC}{RA\ominus}$ is the last one, since $\mathcal{ALC}$ is in monadic $GF^2$, and the role box allows one to express, for example, transitivity. However, the role boxes of $\mathcal{ALC}{RA\ominus}$ can express a lot more than transitivity. Therefore, this result implies the decidability of, e.g. $\mathcal{ALC}{R+}$, but not of $\mathcal{ALC}_{RA\ominus}$. In fact, a much more general result has been shown by Ganzinger et al. (see [7]), but it does not apply to axioms of the form $\forall x, y, z : R(x, y) \land S(y, z) \Rightarrow T(x, z)$.
4 Spatial Reasoning With $\mathcal{ALC}_{RA\ominus}$
A widely accepted approach in the field of spatial reasoning for describing spatial relationships between two-dimensional objects in the plane is to describe their spatial interrelationship qualitatively instead of describing their metrical and/or geometrical attributes. Examples for qualitative spatial calculi fitting into this category are the well-known RCC8 calculus (see [16]) and the so-called Egenhofer-relations (see [6]). In the case of RCC8, we can distinguish 8 disjoint – pairwise exclusive – base relations that describe purely topological aspects of the scene, exhaustively covering the space of all possibilities (see Figure 3). Informally speaking this means that between every two objects in the plane exactly one of the RCC8 relations holds.
Given a set of base relations, e.g. the RCC8 relations, the most important inference problem is the following: given three regions a, b and c in the plane, and the relations R(a, b), S(b, c) between them, what can be deduced about the possible relationships between a and c? This basic inference task is usually given by