Łukasiewicz 神经网络扩展：用于可解释规则提取的残差架构与结晶策略
Łukasiewicz Neural Networks Extended: Residual Architectures and Crystallization Strategies for Interpretable Rule Extraction
AI总结：
本文通过残差连接将Łukasiewicz神经网络扩展到任意深度，并分析三种结晶策略，实现可解释的符号规则提取。
AI中文摘要：
一个权重为整数且激活函数为截断恒等函数的前馈神经网络，逐神经元地实现了 Łukasiewicz 多值逻辑的联结词。这种精确对应关系——由 Castro 和 Trillas 在理论上建立，并由 Leandro 发展为训练算法——使得符号知识提取成为可能：训练产生的不是黑盒模型，而是一个逻辑公式。两个障碍限制了该方法仅适用于浅层架构和小型数据集：结晶（将权重强制为整数）在原始的 Levenberg--Marquardt 训练方案下仅能概率性地成功，而且随着网络加深，理论保证会失效。本文解决了这两个障碍。首先，我们证明了残差连接（ResNet 中使用的那种跳跃连接）将 Łukasiewicz 神经网络扩展到任意深度，同时通过构造在合并神经元处保持符号对应关系：Łukasiewicz 残差块中的合并神经元自动满足神经元分类命题，无论内部层权重如何；内部层神经元通过结晶策略训练以实现可表示性。其次，我们分析了三种结晶策略——Levenberg--Marquardt（修正版）、直通估计器（STE）和近端正则化——刻画了它们的理论保证、失败模式以及可解释性权衡。
英文摘要：
A feed-forward neural network whose weights are integers and whose activation is the truncated identity implements, neuron by neuron, the connectives of Łukasiewicz many-valued logic. This exact correspondence --- established theoretically by Castro and Trillas and developed into a training algorithm by Leandro --- enables \emph{symbolic knowledge extraction}: training produces not a black-box model but a logical formula. Two obstacles have limited the approach to shallow architectures and small datasets: crystallization (forcing weights to integers) succeeds only probabilistically under the original Levenberg--Marquardt training scheme, and the theoretical guarantees break down as networks grow deeper. This paper addresses both obstacles. First, we prove that \emph{residual connections} (skip connections of the kind used in ResNets) extend Łukasiewicz neural networks to arbitrary depth while preserving the symbolic correspondence \emph{at merge neurons} by construction: merge neurons in a Łukasiewicz residual block automatically satisfy the neuron-classification proposition, regardless of the inner layer weights; inner-layer neurons are trained toward representability by the crystallization strategy. Second, we analyse three crystallization strategies --- Levenberg--Marquardt (corrected), straight-through estimation (STE), and proximal regularization --- characterizing their theoretical guarantees, failure modes, and interpretability trade-offs.
