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Three-Dimensional Affine Spatial Logics

Adam Trybus

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.

Three-Dimensional Affine Spatial Logics

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.
Paper Structure (12 sections, 10 theorems, 31 equations, 4 figures)

This paper contains 12 sections, 10 theorems, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. History and philosophy
  3. Constructing a spatial logic
  4. The focus of this article
  5. General setup
  6. Two dimensions
  7. Beyond two dimensions
  8. Three dimensions
  9. Basic expressivity
  10. Addition and multiplication
  11. Affine completeness
  12. Open Problems

Key Result

Proposition 2.2

The set of regular open sets in $X$ forms a Boolean algebra $RO(X)$ with top and bottom defined by $1 = X$ and $0 = \emptyset$, and Boolean operations defined by $a\cdot b = a \cap b$, $a + b = (a \cup b)^{-0}$ and $-a = (X \setminus a)^{0}$.

Figures (4)

  • Figure 1: Example coordinate frames ($m$ is the horizontal and $l$ the vertical line in (a), the image in (b) can be thought of as an affine transformation of that of (a)).
  • Figure 2: A very simple example of Helly's Theorem in $\mathbb{R}^2$.
  • Figure 3: $\overline{\mathbf{OA}} + \overline{\mathbf{OB}} = \overline{\mathbf{OC}}$.
  • Figure 4: $\overline{\mathbf{OA}} \cdot \overline{\mathbf{OB}} = \overline{\mathbf{OC}}$.

Theorems & Definitions (21)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Theorem 3.2: Pratt:1999
  • Theorem 4.1: Helly
  • Theorem 4.2
  • proof
  • ...and 11 more