A Rank-Preserving Locality Theorem

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.