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We rigorously measure the reverse mathematical strengths of fundamental analytic principles for two neural architectures, Transformer and MLP (MultiLayer Perceptron). We prove that the standard compactness principle for Transformer is equivalent (over \(\textsf{RCA}_0\) ) to \(\textsf{ACA}_0\) . For ReLU MLPs on compact domains, we prove that the standard universal approximation theorem for MLPs (with rational parameters) is equivalent to \(\textsf{WKL}_0\) . These results mean that the transformer compactness strictly exceeds the MLP universality in the reverse mathematical hierarchy. This paves the way for a systematic program of reverse mathematical classification of neural networks, bridging proof theory, computable analysis, and machine learning in a novel, interdisciplinary manner.

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Reverse Mathematics for Neural Networks

  • 会议论文

  • 发表日期：

    2026年

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  • Yoshihiro Maruyama

摘要

We rigorously measure the reverse mathematical strengths of fundamental analytic principles for two neural architectures, Transformer and MLP (MultiLayer Perceptron). We prove that the standard compactness principle for Transformer is equivalent (over \(\textsf{RCA}_0\) ) to \(\textsf{ACA}_0\) . For ReLU MLPs on compact domains, we prove that the standard universal approximation theorem for MLPs (with rational parameters) is equivalent to \(\textsf{WKL}_0\) . These results mean that the transformer compactness strictly exceeds the MLP universality in the reverse mathematical hierarchy. This paves the way for a systematic program of reverse mathematical classification of neural networks, bridging proof theory, computable analysis, and machine learning in a novel, interdisciplinary manner.