A Rank-Preserving Gaifman Normal Form for First-Order Logic on Weighted Structures

2026-09-29

AI Summary

This study addresses the complexity and lack of uniform formulation in existing proofs of rank-preserving locality theorems for first-order logic over weighted structures. To this end, it proposes the logic ngFOW+, constructing a semantic framework that integrates techniques from model theory, combinatorics, and algorithmic meta-theory. The primary contributions are twofold. First, it establishes a concise rank-preserving theorem applicable to modular counting quantifiers and the ngFOW+ logic, achieving a canonical form fully consistent with classical Gaifman’s theorem. Second, it simplifies the proof of the near-linear-time decidability algorithm for first-order properties on nowhere dense structures. Collectively, these results lay a solid foundation for future meta-theoretic investigations of this logic.

Abstract

We 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 much simpler and yields formulas in exactly the same normal form as Gaifman's original theorem. Furthermore, it holds not only for first-order logic, but also for first-order logic with modulo-counting quantifiers and, more generally, for the first-order logic on weighted structures ngFOW+ that is introduced in this article. As an application of our theorem, we give a simplified proof of the algorithmic meta-theorem of [Grohe, Kreutzer, Siebertz, JACM 2017] stating that first-order properties of nowhere dense structures can be decided in almost-linear time. Our locality theorem for the weight logic ngFOW+ can be seen as an essential step toward such a meta-theorem for this logic.

Problem

• Gaifman Normal Form
• Rank-Preserving
• First-Order Logic
• Weighted Structures
• Locality Theorem

Innovation

• Rank-Preserving Gaifman Theorem
• Weighted Structures
• First-Order Logic
• Algorithmic Meta-Theorem
• Nowhere Dense