On the decidability of connectedness constraints in 2D and 3D euclidean spaces

Abstract

We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates, as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of reasoning varies dramatically depending on the dimension of the space and on the type of regions considered. For example, the logic with the interior-connectedness predicate (and without contact) is undecidable over polygons or regular closed sets in ℝ2, EXPTIME-complete over polyhedra in ℝ3, and NP-complete over regular closed sets in ℝ3