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

2025-05-12

AI Summary

This work addresses the logical characterization of graph neural networks’ (GNNs) expressive power, aiming to establish precise equivalences between bounded-depth GNN architectures and key fragments of first-order logic (FO). Methodologically, it integrates tools from finite model theory, modal logic, and FO to systematically analyze semantic equivalences between GNNs’ local aggregation mechanisms and the quantifier structures of logical formulas. The paper provides the first bidirectional expressive equivalence proofs linking several logic fragments—including modal logic (ML), graded modal logic (GML), ML with adjacency (ML(A)), two-variable FO (FO₂), and counting two-variable logic (C₂)—with corresponding bounded GNN variants. These results uniformly characterize the expressivity boundaries of diverse GNNs, yielding a tight, formal logical foundation for GNN interpretability. By bridging deep learning theory and classical logic, the work advances the theoretical understanding of GNN capabilities and limitations.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 show that bounded GNN architectures correspond to specific fragments of first-order logic (FO), including modal logic (ML), graded modal logic (GML), modal logic with the universal modality (ML(A)), the two-variable fragment (FO2) and its extension with counting quantifiers (C2). To establish these results, we apply methods and tools from finite model theory of first-order and modal logics to the domain of graph representation learning. This provides a unifying framework for understanding the logical expressiveness of GNNs within FO.

Problem

• Understanding expressive power of bounded GNNs
• Mapping GNNs to first-order logic fragments
• Unifying logical expressiveness of GNNs

Innovation

• Bounded GNNs correspond to first-order logic fragments
• Uses finite model theory for graph representation analysis
• Links GNN expressiveness to modal and counting logics