Three-Dimensional Affine Spatial Logics

TL;DR

This paper investigates three-dimensional affine spatial logics, demonstrating their expressive power and establishing that regions satisfying certain formulas are affine equivalent, thus advancing the understanding of affine geometry in spatial reasoning.

Contribution

It shows that different dimensional affine logics have distinct theories and constructs formulas to describe 3D coordinate frames, highlighting the logic's high expressive power.

Findings

Different dimensional affine logics have distinct theories.

Formulas can describe three-dimensional coordinate frames.

Every region satisfying an affine complete formula is affine equivalent.

Abstract

We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to analyse certain family of affine logics with inclusion and convexity as primitives interpreted over real spaces of increasing dimensionality. In this article we show that logics of different dimensionalities must have different theories, thus justifying further work on different dimensions. We then focus on the three-dimensional case, exploring the expressiveness of this logic and consequently showing that it is possible to construct formulas describing a three-dimensional coordinate frame. The final result, making use of the high expressive power of this logic, is that every region satisfies an affine complete formula, meaning that all regions satisfying it are affine equivalent.