Topological Logics with Connectedness over Euclidean Spaces
We consider the quantifier-free languages, Bc and Bc(omicron), 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 R-n (n >= 2) and, additionally, over the regular closed semilinear sets of R-n. 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 semilinear sets in all dimensions greater than 1, and that the satisfiability problem for Bc and Bc(omicron) is undecidable over both the regular closed sets and the regular closed semilinear sets in the Euclidean plane. However, we also prove that the satisfiability problem for Bc(omicron) is NP-complete over the regular closed sets in all dimensions greater than 2, while the corresponding problem for the regular closed semilinear sets is EXPTIME-complete. Our results show, in particular, that spatial reasoning is much harder over Euclidean spaces than over arbitrary topological spaces.
topology connectedness euclidean spaces qualitative spatial reasoning theory