A Rank-Preserving Gaifman Normal Form

Martin Grohe; Nicole Schweikardt

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arXiv · 2606.11993

A Rank-Preserving Gaifman Normal Form

Martin Grohe·Nicole Schweikardt

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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.

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Martin Grohe, Nicole Schweikardt. 2026-06-10. A Rank-Preserving Gaifman Normal Form. https://arxiv.org/abs/2606.11993

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