Structural Preservation and the Logical Expressiveness of Graph Neural Networks

2026-06-16

AI Summary

This work investigates the logical expressiveness of graph neural networks (GNNs) under structure-preserving mappings—specifically embeddings, injective homomorphisms, and general homomorphisms. By integrating graded modal logic, graph homomorphism theory, and novel well-quasi-ordering techniques for trees of bounded height, the paper establishes, for the first time and without reliance on any specific GNN architecture, a correspondence between these structural preservation properties and fragments of modal logic. The main contributions include a new well-quasi-ordering result for trees, a framework for translating between GNNs and logical formulas, and proofs that the three classes of structure-preserving mappings are respectively equivalent to existential graded modal logic, its positive existential fragment, and positive existential modal logic. These equivalences directly yield GNN architectures with matching expressive power.

Abstract

Bridges between graph neural networks (GNNs) and logical formalisms have been established by fixing architectural choices, such as the types of aggregation, combination, and activation functions. These choices define restricted classes of GNNs for which tight correspondences with logical formalisms can be obtained, by showing that logical formulae can be translated into equivalent GNNs and, conversely, that GNNs can be translated into equivalent formulae. In this paper we take a semantic perspective by establishing the logical expressiveness of classes of GNN classifiers that are preserved under structural properties: embeddings (extensions), injective homomorphisms, and homomorphisms. We show that, for each such property, there exists a fragment of graded modal logic characterising the class of GNNs. In particular, preservation under embeddings, injective homomorphisms, and homomorphisms corresponds to existential graded modal logic, its existential-positive fragment, and existential-positive modal logic, respectively. These results characterise the expressiveness of broad classes of GNNs independently of specific architectural choices, but we also show that each of these classes admits a GNN architecture of the same expressiveness. Technically, our approach uses a new well-quasi-order result for trees of bounded height, yielding finite representations of unravelling-invariant classes.

Problem

• graph neural networks
• logical expressiveness
• structural preservation
• modal logic
• homomorphisms

Innovation

• logical expressiveness
• structural preservation
• graded modal logic
• graph neural networks
• well-quasi-order