Rank-preserving locality theorem simplifies logic analysis on weighted structures
A Rank-Preserving Gaifman Normal Form for First-Order Logic on Weighted Structures
Logic in Computer Science
Summary
The paper presents a simplified and more general version of Gaifman's Theorem, which helps break down complex logical statements into local parts without increasing their complexity. The authors extend this result beyond ordinary first-order logic to more expressive logical systems that include counting and weighted elements. This new theorem aids in understanding and deciding logical properties efficiently on large but sparse data structures. It also provides a clearer path toward faster algorithms for complex logical queries in these settings.
What this means in practice
- •For database engineers: Develop query optimization methods that maintain formula complexity on sparse weighted data graphs.
- •For network protocol designers: Use locality properties to efficiently verify logical conditions in distributed network topologies with weighted connections.
A theory result. No direct application yet.
Authors
Steffen van Bergerem, Martin Grohe, Charlotte Lenz, Nicole Schweikardt
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.