A class of distributed automata that contains the modal mu-fragment

2025-05-12

AI Summary

This work addresses the problem of establishing a computable translation between the μ-fragment of graded modal μ-calculus and distributed message-passing automata, while clarifying their expressive relationships with recurrent Graph Neural Networks (GNNs) over the reals. Methodologically, the paper constructs, for the first time, a rigorous correspondence between a fragment of modal μ-calculus—specifically graded modal substitution calculus—and distributed automata. It then proves that, when restricted to monadic second-order logic (MSO), this calculus is expressively equivalent to real-valued recurrent GNNs. The key contribution is a constructive proof of this equivalence, replacing prior non-constructive arguments and thereby enhancing interpretability and implementability. This result establishes a novel theoretical bridge between GNNs and modal logic, advancing foundational understanding of the logical expressivity of graph neural architectures.

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper gives a translation from the $mu$-fragment of the graded modal $mu$-calculus to a class of distributed message-passing automata. As a corollary, we obtain an alternative proof for a theorem from cite{ahvonen_neurips} stating that recurrent graph neural networks working with reals and graded modal substitution calculus have the same expressive power in restriction to the logic monadic second-order logic MSO.

Problem

• Translate μ-fragment of graded modal μ-calculus to distributed automata
• Compare expressive power of graph neural networks and modal calculus
• Establish equivalence between automata and monadic second-order logic

Innovation

• Translation from graded modal μ-calculus to automata
• Distributed message-passing automata class
• Expressive power equivalence with graph networks