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通过卢卡谢维奇逻辑完全识别深层ReLU网络

文章背景与核心概要

本文探讨了深层神经网络中函数非唯一性的根本挑战——具体而言，具有完全不同架构和参数的两个深层ReLU网络如何能够实现完全相同的函数。

受克劳德·香农（Claude Shannon）利用布尔逻辑对开关电路进行历史性分析的启发，作者为深层ReLU网络开发了一种符号演算方法。通过将网络识别映射到通过多值（MV）逻辑公理推导卢卡谢维奇公式（Łukasiewicz formulae），该框架刻画了ReLU网络的所有架构对称性。

所提出的方法由三个核心步骤组成： 1. 提取算法： 将神经网络转换为分层的替代图（substitution graph），其真值函数对应于网络的输入-输出映射。 2. 完备性定理： 证明了功能等效的公式可以使用特定的MV逻辑公理（整数、有理数或实数权重和偏置）相互推导。 3. 构建算法： 从替代图反向重建神经网络。

最终，该方法引入了一种MV逻辑的组合范式（compositional normal form），通过节点重写、层坍缩和层扩展这三种局部操作，保持了深层神经网络的代数结构。

摘要 (Summary)

This paper addresses the fundamental challenge of functional non-uniqueness in deep neural networks—specifically, how two deep ReLU networks with entirely different architectures and parameters can realize the exact same function.

Drawing inspiration from Claude Shannon’s historical analysis of switching circuits using Boolean logic, the authors develop a symbolic calculus for deep ReLU networks. By mapping network identification to the derivation of Łukasiewicz formulae via many-valued (MV) logic axioms, the framework characterizes all architectural symmetries of ReLU networks.

The proposed methodology consists of three core steps: 1. Extraction Algorithm: Transforms a neural network into a layered substitution graph, whose truth function corresponds to the network's input-output map. 2. Completeness Theorem: Proves that functionally equivalent formulae are interderivable using specific MV logic axioms (integer, rational, or real weights and biases). 3. Construction Algorithm: Reconstructs the neural network back from the substitution graph.

Ultimately, this approach introduces a compositional normal form for MV logic that preserves the algebraic structure of deep neural networks through three local operations: node rewrite, layer collapse, and layer expansion.

文章元数据 (Article Metadata)

  • arXiv Identifier: arXiv:2602.00266
  • Primary Subject: Artificial Intelligence (cs.AI)
  • Authors: Yani Zhang, Helmut Bölcskei
  • Submitted: January 30, 2026
  • Last Revised: September 3, 2026 (Version v2)
  • DOI: 10.48550/arXiv.2602.00266

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