Three-Dimensional Affine Spatial Logics
This work addresses a foundational problem in spatial logic for researchers in geometry and logic, but it is incremental as it extends prior work from two to three dimensions.
The paper tackles the problem of analyzing affine spatial logics in three-dimensional real spaces, showing that logics of different dimensionalities have distinct theories and constructing formulas to describe a three-dimensional coordinate frame, with the result that every region satisfies an affine complete formula, implying all such regions are affine equivalent.
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.