Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces

The language RCC8 is a widely-studied formalism for describing topological arrangements of spatial regions. The variables of this language range over the collection of non-empty, regular closed sets of n-dimensional Euclidean space, here denoted RC(Rn), and its non-logical primitives allow us to specify how the interiors, exteriors and boundaries of these sets intersect. The key question is the satisfiability problem: given a finite set of atomic RCC8constraints in m variables, determine whether there exists an m-tuple of elements of RC(Rn) satisfying them. These problems are known to coincide for all n ≥ 1, so that RCC8-satisfiability is independent of dimension. This common satisfiability problem is NLogSpace-complete. Unfortunately, RCC8 lacks the means to say that a spatial region comprises a ‘single piece’, and the present article investigates what happens when this facility is added. We consider two extensions of RCC8: RCC8c, in which we can state that a region is connected, and RCC8c◦, in which we can instead state that a region has a connected interior. The satisfiability problems for both these languages are easily seen to depend on the dimension n, for n ≤ 3. Furthermore, in the case of RCC8c◦, we show that there exist finite sets of constraints that are satisfiable over RC(R2), but only by ‘wild’ regions having no possible physical meaning. This prompts us to consider interpretations over the more restrictive domain of non-empty, regular closed, polyhedral sets, RCP(Rn). We show that (a) the satisfiability problems for RCC8c (equivalently, RCC8c◦) over RC(R) and RCP(R) are distinct and both NP-complete; (b) the satisfiability problems for RCC8c over RC(R2) and RCP(R2) are identical and NP-complete; (c) the satisfiability problems for RCC8c◦ over RC(R2) and RCP(R2) are distinct, and the latter is NP-complete. Decidability of the satisfiability problem for RCC8c◦ over RC(R2) is open. For n ≥ 3, RCC8c and RCC8c◦ are not interestingly different from RCC8. We finish by answering the following question: given that a set of RCC8cor RCC8c◦-constraints is satisfiable over RC(Rn) or RCP(Rn), how complex is the simplest satisfying assignment? In particular, we exhibit, for both languages, a sequence of constraints Φn, satisfiable over RCP(R2), such that the size of Φn grows polynomially in n, while the smallest configuration of polygons satisfying Φn cuts the plane into a number of pieces that grows exponentially. We further show that, over RC(R2), RCC8c again requires exponentially large satisfying diagrams, while RCC8c◦ can force regions in satisfying configurations to have infinitely many components.