The Correspondence Between Bounded Graph Neural Networks and Fragments of First-Order Logic

TL;DR

This paper establishes a precise correspondence between certain GNN architectures and fragments of first-order logic, providing a theoretical framework to understand their expressive power in graph representation learning.

Contribution

It introduces GNN models aligned with specific FO logic fragments, bridging graph neural networks with formal logical expressiveness using finite model theory.

Findings

GNNs correspond exactly to certain FO logic fragments.

Provides a unifying logical framework for GNN expressiveness.

Uses finite model theory to analyze GNN capabilities.

Abstract

Graph Neural Networks (GNNs) address two key challenges in applying deep learning to graph-structured data: they handle varying size input graphs and ensure invariance under graph isomorphism. While GNNs have demonstrated broad applicability, understanding their expressive power remains an important question. In this paper, we propose GNN architectures that correspond precisely to prominent fragments of first-order logic (FO), including various modal logics as well as more expressive two-variable fragments. To establish these results, we apply methods from finite model theory of first-order and modal logics to the domain of graph representation learning. Our results provide a unifying framework for understanding the logical expressiveness of GNNs within FO.