A Rank-Preserving Gaifman Normal Form

2026-06-10

AI Summary

This work aims to simplify the application of locality theorems for first-order logic while preserving the rank of formulas. To this end, we introduce a novel rank measure and establish a rank-preserving variant of Gaifman’s theorem, yielding formulas that strictly retain the structural properties of the original Gaifman normal form. Compared to existing approaches, our proof substantially streamlines the derivation of rank-preserving locality results. As an application, we provide a concise proof of the important result that first-order properties are decidable in near-linear time on nowhere-dense graph classes, thereby demonstrating both the efficacy and theoretical significance of our method.

Abstract

We introduce a rank measure for first-order logic and prove a "rank-preserving'" version of Gaifman's theorem. Compared to earlier "rank-preserving locality theorems'" (in particular, [Grohe, Kreutzer, Siebertz, JACM 2017]), our theorem is not only much simpler, but also yields formulas in exactly the same normal form as Gaifman's original theorem. As an application of this theorem, we give a simplified proof of the main result of [Grohe, Kreutzer, Siebertz, JACM 2017] that first-order properties of nowhere-dense structures can be decided in almost linear time.

Problem

• first-order logic
• Gaifman normal form
• rank-preserving
• nowhere-dense structures
• locality theorems

Innovation

• rank-preserving
• Gaifman normal form
• first-order logic
• nowhere-dense structures
• locality theorem