Aggregate-Combine-Readout GNNs Are More Expressive Than Logic C2

2025-08-08

AI Summary

This paper resolves an open question posed by Barceló et al. (2020): whether aggregate-combine-readout graph neural networks (GNNs) possess strictly greater logical expressivity than the two-variable counting logic C². Method: Leveraging a novel correspondence between GNNs and infinitary logic, the authors combine structural isomorphism analysis with tools from model theory to characterize the distinguishing power of readout-equipped GNNs. Results: The work provides the first rigorous proof that aggregate-combine-readout GNNs—equipped with readout mechanisms—can distinguish graph structures indistinguishable in C², both on directed and undirected graphs. Consequently, their expressive power strictly exceeds that of full C². This result settles a long-standing theoretical open problem and establishes a new upper bound on the logical expressivity of this prominent GNN architecture, thereby strengthening foundational connections between graph representation learning and mathematical logic.

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 learning to rank, online learning, and counterfactual learning for ranking

Abstract

In recent years, there has been growing interest in understanding the expressive power of graph neural networks (GNNs) by relating them to logical languages. This research has been been initialised by an influential result of Barceló et al. (2020), who showed that the graded modal logic (or a guarded fragment of the logic C2), characterises the logical expressiveness of aggregate-combine GNNs. As a ``challenging open problem'' they left the question whether full C2 characterises the logical expressiveness of aggregate-combine-readout GNNs. This question has remained unresolved despite several attempts. In this paper, we solve the above open problem by proving that the logical expressiveness of aggregate-combine-readout GNNs strictly exceeds that of C2. This result holds over both undirected and directed graphs. Beyond its implications for GNNs, our work also leads to purely logical insights on the expressive power of infinitary logics.

Problem

• Comparing expressive power of GNNs and logic C2
• Resolving open problem on GNNs exceeding C2 expressiveness
• Analyzing GNNs' logical expressiveness over undirected and directed graphs

Innovation

• Aggregate-combine-readout GNNs exceed C2 expressiveness
• Proven over undirected and directed graphs
• Provides insights into infinitary logics