Axiomatizing rectangular grids with no extra non-unary relations

TL;DR

This paper presents a formula that axiomatizes non-narrow rectangular grids using only grid neighborship relations, and explores the spectrum of formulas with planar models related to computational complexity.

Contribution

It introduces a new axiomatization of rectangular grids without binary relations beyond neighborship, linking model spectra to Turing machine recognizability within specific time-space bounds.

Findings

A formula axiomatizes non-narrow rectangular grids using only neighborship relations.

Spectra of formulas with planar models correspond to sets recognized by Turing machines within certain time-space constraints.

The paper establishes a connection between logical definability and computational complexity for planar models.

Abstract

We construct a formula $\phi$ which axiomatizes non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set $A \subseteq \mathbb{N}$ is a spectrum of a formula which has only planar models if numbers $n \in A$ can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time $t(n)$ and space $s(n)$, where $t(n)s(n) \leq n$ and $t(n),s(n) = \Omega(\log(n))$.