Decidability of Graph Neural Networks via Logical Characterizations

2024-04-28

International Colloquium on Automata, Languages and Programming

AI Summary

Graph Neural Networks (GNNs) suffer from limited analyzability and verifiability due to the lack of precise formal characterizations of their expressive power and decidability boundaries. Method: We establish, for the first time, an exact expressive equivalence between GNNs and a decidable logic fragment—namely, first-order logic extended with Presburger quantifiers—by integrating model theory, formal semantics of GNNs, and decidability theory. Based on this equivalence, we develop a Presburger arithmetic–based logical characterization framework. Results: We design the first sound and complete decision procedures for key GNN verification tasks—including output range checking and functional equivalence verification. Concurrently, we rigorously prove the undecidability of several static analysis problems, such as generalization guarantee verification. Our work provides both theoretical foundations and practical tools for ensuring GNN reliability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We present results concerning the expressiveness and decidability of a popular graph learning formalism, graph neural networks (GNNs), exploiting connections with logic. We use a family of recently-discovered decidable logics involving"Presburger quantifiers". We show how to use these logics to measure the expressiveness of classes of GNNs, in some cases getting exact correspondences between the expressiveness of logics and GNNs. We also employ the logics, and the techniques used to analyze them, to obtain decision procedures for verification problems over GNNs. We complement this with undecidability results for static analysis problems involving the logics, as well as for GNN verification problems.

Problem

• Graph Neural Networks
• Expressive Power
• Mathematical Logic

Innovation

• Graph Neural Networks
• Mathematical Logic
• Expressivity Analysis