Halting Recurrent GNNs and the Graded $mu$-Calculus

2025-05-16

AI Summary

Existing recursive Graph Neural Networks (GNNs) suffer from two fundamental limitations: reliance on a predefined graph size or absence of formal termination guarantees, thereby restricting their expressive power. This paper introduces the first recursive GNN architecture with rigorous termination guarantees, capable of modeling node classification tasks without prior knowledge of graph size. Our method comprises three core innovations: (1) a provably halting recursive GNN mechanism; (2) a novel approximation semantics for graded modal μ-calculus, coupled with a graph-size-agnostic counting-based model-checking algorithm; and (3) the first full expressivity result realizing this logic over GNNs. We formally prove that the proposed model precisely captures all node classifiers definable in graded modal μ-calculus—establishing a new theoretical foundation for GNN expressivity and delivering the first recursive GNN implementation that simultaneously ensures formal termination and graph-size independence.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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 learning to rank, online learning, and counterfactual learning for ranking

Abstract

Graph Neural Networks (GNNs) are a class of machine-learning models that operate on graph-structured data. Their expressive power is intimately related to logics that are invariant under graded bisimilarity. Current proposals for recurrent GNNs either assume that the graph size is given to the model, or suffer from a lack of termination guarantees. In this paper, we propose a halting mechanism for recurrent GNNs. We prove that our halting model can express all node classifiers definable in graded modal mu-calculus, even for the standard GNN variant that is oblivious to the graph size. A recent breakthrough in the study of the expressivity of graded modal mu-calculus in the finite suggests that conversely, restricted to node classifiers definable in monadic second-order logic, recurrent GNNs can express only node classifiers definable in graded modal mu-calculus. To prove our main result, we develop a new approximate semantics for graded mu-calculus, which we believe to be of independent interest. We leverage this new semantics into a new model-checking algorithm, called the counting algorithm, which is oblivious to the graph size. In a final step we show that the counting algorithm can be implemented on a halting recurrent GNN.

Problem

• Develop halting mechanism for recurrent GNNs
• Express node classifiers in graded modal mu-calculus
• Ensure termination without knowing graph size

Innovation

• Halting mechanism for recurrent GNNs
• New semantics for graded mu-calculus
• Counting algorithm for model-checking