用多值逻辑破解深度ReLU网络的函数对称性，实现完全识别
Complete Identification of Deep ReLU Neural Networks by Many-Valued Logic
- 将ReLU网络映射为卢卡西维茨逻辑公式，通过代数重写实现等效变换
- 证明每个函数等价类中的网络可通过有限对称性连接，且数量有限
- 方法类似香农电路设计，适合研究网络可识别性与结构等价问题
深度ReLU神经网络存在非平凡的函数对称性：结构和参数迥异的网络可能实现相同函数。本文解决完全识别问题——给定函数f，推导出所有产生该函数的前馈ReLU网络的架构与参数。将ReLU网络转化为卢卡西维茨逻辑公式，利用逻辑公理进行代数重写以实现函数等价的网络变换。提出一种组合范式，便于从卢卡西维茨逻辑公式还原为ReLU网络。基于Chang完备性定理，证明每个函数等价类中所有ReLU网络均通过一组有限对称性相连，而这些对称性对应于卢卡西维茨逻辑的有限公理集。该思想类似于香农关于开关电路设计的经典工作，即电路被转换为布尔公式，合成过程由布尔逻辑公理指导的代数重写实现。
原文摘要 · Abstract (English)
Deep ReLU neural networks admit nontrivial functional symmetries: vastly different architectures and parameters (weights and biases) can realize the same function. We address the complete identification problem -- given a function f, deriving the architecture and parameters of all feedforward ReLU networks giving rise to f. We translate ReLU networks into Lukasiewicz logic formulae, and effect functional equivalent network transformations through algebraic rewrites governed by the logic axioms. A compositional norm form is proposed to facilitate the mapping from Lukasiewicz logic formulae back to ReLU networks. Using Chang's completeness theorem, we show that for every functional equivalence class, all ReLU networks in that class are connected by a finite set of symmetries corresponding to the finite set of axioms of Lukasiewicz logic. This idea is reminiscent of Shannon's seminal work on switching circuit design, where the circuits are translated into Boolean formulae, and synthesis is effected by algebraic rewriting governed by Boolean logic axioms.
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