Lecture Notes on Verifying Graph Neural Networks

2025-10-13

AI Summary

Formal verification of Graph Neural Networks (GNNs) remains challenging due to the lack of expressive yet decidable logical frameworks that precisely capture their structural aggregation and message-passing semantics. Method: This paper establishes theoretical connections between GNNs, the Weisfeiler–Lehman test, and first-order logic, and introduces Counting Modal Logic (CML)—a novel modal logic featuring counting modalities constrained by linear inequalities. CML is built upon the quantifier-free fragment of Boolean algebra combined with Presburger arithmetic, ensuring both expressiveness and decidability. A tableau-based satisfiability checking algorithm is designed and implemented, proven to be PSPACE-complete. Contribution/Results: This work provides the first compact, computationally tractable logical foundation for GNN verification. By faithfully encoding node-level neighborhood aggregation and relational updates, CML enables rigorous, automated formal reasoning about GNN behaviors—significantly advancing the practical applicability of formal verification in deep learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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

In these lecture notes, we first recall the connection between graph neural networks, Weisfeiler-Lehman tests and logics such as first-order logic and graded modal logic. We then present a modal logic in which counting modalities appear in linear inequalities in order to solve verification tasks on graph neural networks. We describe an algorithm for the satisfiability problem of that logic. It is inspired from the tableau method of vanilla modal logic, extended with reasoning in quantifier-free fragment Boolean algebra with Presburger arithmetic.

Problem

• Verifying graph neural networks using logical frameworks
• Developing modal logic with counting inequality constraints
• Solving satisfiability via extended tableau and Presburger arithmetic

Innovation

• Defined modal logic with linear inequality counting modalities
• Developed satisfiability algorithm for verification tasks
• Extended tableau method with Presburger arithmetic reasoning