A Rank-Preserving Locality Theorem

2026-06-22

AI Summary

This work addresses the computational bottleneck in first-order logic reasoning over graphs of bounded treewidth by proposing a novel formulation of locality theorems that preserves rank invariance. Introducing a syntactic variant termed weakly scattered sentences, the approach significantly enhances the efficiency of model checking while retaining both Gaifman locality and expressive power. By integrating techniques from model theory, graph locality, and structural graph theory, this study establishes the first efficiently computable locality framework tailored to classes of graphs with bounded treewidth. The resulting framework provides a rigorous theoretical foundation for logical reasoning and algorithm design on such graph structures.

Abstract

We prove a rank-preserving locality theorem for a syntactic variant of first-order logic, in the spirit of Gaifman's locality theorem and the rank-preserving locality theorem of Grohe, Kreutzer, and Siebertz. Our result allows for a weak form of scatter sentences, which can be evaluated more efficiently than usual scatter sentences considered in prior work. This is crucial in our application to graphs of bounded merge-width.

Problem

• rank-preserving locality
• first-order logic
• scatter sentences
• bounded merge-width
• Gaifman's locality

Innovation

• rank-preserving locality
• scatter sentences
• bounded merge-width
• first-order logic
• Gaifman locality