On the Computability of Region-Based Euclidean Logics

Abstract

AbstractBy 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 higher-dimensional 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.KeywordsTopological SpaceContact RelationTernary RelationSpatial LogicContact AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.