A Logic of Strong Contact between Polytopes

TL;DR

This paper introduces a new strong contact relation between polytopes, proves its properties, and explores the logical systems describing these contact relations, showing their equivalence with standard connected contact algebra logics.

Contribution

It defines and analyzes a novel strong contact relation for polytopes and characterizes its logical properties and equivalences with existing contact algebra systems.

Findings

Strong contact relation is a contact relation for polytopes.

Universal fragments of the logic coincide with standard quantifier-free systems.

Results unify various classes of contact algebras.

Abstract

We propose a new contact relation between polytopes. Intuitively, we say that two polytopes are in strong contact if a small enough object can pass from one of them to the other while remaining in their union. In the first half of the paper we prove that this relation is indeed a contact relation between polytopes, which turns out not to be the case for arbitrary regular closed in Euclidean spaces sets. In the second half we study the universal fragments of the logics of the resultant contact algebras. We prove that they all coincide with the set of theorems of a standard quantifier-free formal system for connected contact algebras, which also coincides with the universal fragments of the logics of a variety of (classes of) contact algebras of interest.