Abstract
Graph neural networks (GNNs) have gained widespread application in various real-world scenarios due to their powerful ability to handle graph-structured data. However, the computational power and logical expressiveness of GNNs are still not fully understood. This work explores the logical expressiveness of GNNs from a theoretical view and establishes a connection between them and the fragment of first-order logic, known as \(\mathcal {C}_2\), which servers as a logical language for graph data modeling. A recent study proposes a type of GNNs called ACR-GNN, demonstrating that GNNs can mimic the evaluation of unary \(\mathcal {C}_2\) formulas. Working upon this, we give a variant of GNN architectures capable of handling general \(\mathcal {C}_2\) formulas. To achieve this, we leverage a mechanism known as message passing to reconstruct GNNs. The proposed GNN variants enable the simultaneous updating of node and node pair features, allowing for the handling of both unary and binary predicates in \(\mathcal {C}_2\) formulas. We prove that the proposed models possess the same expressiveness as \(\mathcal {C}_2\). Through experiments conducted on synthetic and real datasets, we validate that our proposed models outperform both ACR-GNN and a widely-used model, GIN, in the tasks of evaluating \(\mathcal {C}_2\) formulas.
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Notes
- 1.
A variable x is free if it is not qualified by \(\exists \) and \(\forall \). See the formula \(\exists y.\varphi (x,y)\), where variable x is free and variable y is not.
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Acknowledgement
Zhangquan Zhou and Shijiao Tang were supported by The Natural Science Foundation of the Jiangsu Higher Education Institutions of China under grant 22KJB520003. Qianqian Zhang was supported by the same foundation under grant 22KJD510009.
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Zhou, Z., Yang, C., Zhang, Q., Tang, S. (2023). Exploring the Logical Expressiveness of Graph Neural Networks by Establishing a Connection with \(\mathcal {C}_2\). In: Wang, H., Han, X., Liu, M., Cheng, G., Liu, Y., Zhang, N. (eds) Knowledge Graph and Semantic Computing: Knowledge Graph Empowers Artificial General Intelligence. CCKS 2023. Communications in Computer and Information Science, vol 1923. Springer, Singapore. https://doi.org/10.1007/978-981-99-7224-1_3
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