This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of computational complexity. In particular, we draw attention to the special problems which arise when the logics are interpreted not over arbitrary topological spaces, but over (low-dimensional) Euclidean spaces.

The main result of this paper is to show that the problem of instantiating a finite and path-consistent constraint network of lines in the Euclidean space is NP-complete. Indeed, we already know that reasoning with lines in the Euclidean space is NPhard. In order to prove that this problem is NPcomplete, we first establish that a particular instance of this problem can be solved by a nondeterministic polynomial-time algorithm, and then we show that solving any finite and path-consistent constraint network of lines in the Euclidean space is at most as difficult as solving that instance. Keywords. Constraint satisfaction problems, Satisfiability, Qualitative spatial reasoning, Euclidean geometry.

Abstract. By a Euclidean logic, we understand a formal language whose variables range over subsets of Euclidean space, of some fixed dimension, and whose non-logical primitives have fixed meanings as geometrical properties, relations and operations involving those sets. In this paper, we consider first-order Euclidean logics with primitives for the properties of connectedness and convexity, the binary relation of contact and the ternary relation of being closer-than. We investigate the computational properties of the corresponding first-order theories when variables are taken to range over various collections of subsets of 1-, 2- and 3-dimensional space. We show that the theories based on Euclidean spaces of dimension greater than 1 can all encode either first- or second-order arithmetic, and hence are undecidable. We show that, for logics able to express the closer-than relation, the theories of structures based on 1-dimensional Euclidean space have the same complexities as their higherdimensional counterparts. By contrast, in the absence of the closer-than predicate, all of the theories based on 1-dimensional Euclidean space considered here are decidable, but non-elementary. 1