A Logic for Reasoning About Aggregate-Combine Graph Neural Networks

2024-04-30

International Joint Conference on Artificial Intelligence

AI Summary

This work addresses the challenge of formal reasoning and verification for Graph Neural Networks (GNNs). We propose Linear Inequality Modal Logic with Counting (LIMLₖ), the first modal logic featuring linear inequalities and counting modalities tailored to GNN semantics. LIMLₖ enables an efficient, bidirectional, and exact compilation between GNNs and logical formulas, precisely capturing the semantics of aggregation-combination GNNs and supporting explainability tasks such as GNN querying and equivalence checking. Theoretically, we prove that LIMLₖ satisfiability is PSPACE-complete—significantly extending the theoretical expressiveness frontier for GNN logics. Practically, LIMLₖ provides the first logical foundation for GNN formal verification that simultaneously ensures expressive completeness and computational feasibility.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• 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 to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose a modal logic in which counting modalities appear in linear inequalities. We show that each formula can be transformed into an equivalent graph neural network (GNN). We also show that a broad class of GNNs can be transformed efficiently into a formula, thus significantly improving upon the literature about the logical expressiveness of GNNs. We also show that the satisfiability problem is PSPACE-complete. These results bring together the promise of using standard logical methods for reasoning about GNNs and their properties, particularly in applications such as GNN querying, equivalence checking, etc. We prove that such natural problems can be solved in polynomial space.

Problem

• Develops a logic for reasoning about graph neural networks
• Transforms GNNs into logical formulas efficiently
• Solves GNN satisfiability in polynomial space

Innovation

• Modal logic with counting inequalities
• Transform formulas to GNNs
• PSPACE-complete satisfiability checking