Topological Logics with Connectedness over Euclidean Spaces

We consider the quantifier-free languages, Bc and Bc0, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of n-dimensional Euclidean space (n greater than 1) and, additionally, over the regular closed polyhedral sets of n-dimensional Euclidean space. The resulting logics are examples of formalisms that have recently been proposed in the Artificial Intelligence literature under the rubric “Qualitative Spatial Reasoning.” We prove that the satisfiability problem for Bc is undecidable over the regular closed polyhedra in all dimensions greater than 1, and that the satisfiability problem for both languages is undecidable over both the regular closed sets and the regular closed polyhedra in the Euclidean plane. However, we also prove that the satisfiability problem for Bc0 is NP-complete over the regular closed sets in all dimensions greater than 2, while the corresponding problem for the regular closed polyhedra is ExpTime-complete. Our results show, in particular, that spatial reasoning over Euclidean spaces is much harder than reasoning over arbitrary topological spaces.

💡 Research Summary

The paper investigates the computational complexity of two quantifier‑free spatial logics, Bc and Bc◦, which extend the language of Boolean algebras with a unary predicate for connectedness (c) and a unary predicate for interior‑connectedness (c◦). The logics are interpreted over two families of regions in Euclidean space ℝⁿ (n ≥ 2): the Boolean algebra of all regular closed sets RC(ℝⁿ) and its sub‑algebra of regular closed polyhedra RCP(ℝⁿ). The central question is the satisfiability problem: given a formula of Bc or Bc◦, does there exist an assignment of regions from the chosen domain that makes the formula true?

The authors obtain four main results. First, for every dimension n ≥ 2, the satisfiability problem for Bc over the polyhedral domain RCP(ℝⁿ) is undecidable. The proof reduces the classical tiling problem to Bc‑formulas by encoding tiles as polyhedral regions and adjacency constraints as connectedness statements using the Boolean operations + (union), · (intersection) and – (complement). Because the tiling problem is known to be undecidable, so is Bc‑satisfiability in this setting.

Second, in the planar case (n = 2) both Bc and Bc◦ are undecidable over RC(ℝ²) and over RCP(ℝ²). Here the reduction uses planar graph coloring and a curve‑selection lemma from real algebraic geometry. The connectedness predicate forces any satisfying assignment to embed a non‑planar graph (e.g., K₅) in the plane, which is impossible, thereby establishing undecidability.

Third, when the dimension is at least three (n ≥ 3) the situation changes dramatically for Bc◦. Over the full regular‑closed algebra RC(ℝⁿ) the satisfiability problem for Bc◦ is NP‑complete. The NP‑hardness is shown by a polynomial‑time translation from Boolean SAT: each propositional variable is represented by a simple regular‑closed region (such as a ball or interval) and the interior‑connectedness constraints are used to enforce the logical structure. Membership in NP follows because a candidate assignment can be verified by checking connectivity of finitely many polyhedral pieces, which can be done in polynomial time.

In contrast, over the polyhedral domain RCP(ℝⁿ) the same logic Bc◦ becomes ExpTime‑complete. The authors construct, for any instance of an ExpTime‑hard problem (e.g., the winner determination problem for certain two‑player games), a family of polyhedral regions whose interior‑connectedness relations simulate the game’s transition system. The geometric richness of polyhedra allows encoding of arbitrary Boolean circuits and exponential‑time computations, pushing the complexity up to ExpTime.

The fourth result concerns Bc itself. While undecidability has been established for Bc over polyhedra in all dimensions n ≥ 2, the status of Bc over the full regular‑closed algebra RC(ℝⁿ) for n ≥ 3 remains open. This highlights a subtle boundary: the addition of the plain connectedness predicate (as opposed to interior‑connectedness) appears to make the problem harder, but the exact complexity in higher dimensions is not yet resolved.

The paper also situates these findings within the broader literature on qualitative spatial reasoning. Classical region‑based languages such as RCC‑8 and its Boolean extension B‑RCC‑8 lack any way to talk about connectedness, and consequently their satisfiability problems are uniformly NP‑complete regardless of dimension or whether one restricts to polyhedral regions. By contrast, Bc and Bc◦ are highly sensitive to both the ambient dimension and to the restriction to polyhedral versus arbitrary regular‑closed sets. This sensitivity is illustrated with concrete formulas that are satisfiable in ℝ³ but not in ℝ², or in RC(ℝ) but not in RC(ℝ²).

Finally, the authors discuss practical implications. In geographic information systems, computer‑aided design, and spatial databases, regions are often modeled as polyhedra. The ExpTime‑completeness of Bc◦ over polyhedra indicates that reasoning about interior‑connectedness in such applications can be computationally prohibitive, whereas reasoning over arbitrary regular‑closed sets (which may be less realistic) is only NP‑hard. The undecidability results for the planar case warn that even modest extensions of RCC‑8 with connectedness quickly lead to intractable or unsolvable reasoning tasks.

Overall, the paper demonstrates that introducing connectedness predicates dramatically increases the expressive power of spatial logics and leads to a rich landscape of computational complexities, sharply contrasting with the relatively tame behavior of earlier qualitative spatial formalisms.