将神经网络与模态逻辑的融合方法建立对应关系，揭示深层学习模型的逻辑表达能力。
From Neural Networks to Logical Theories: The Correspondence between Fibring Modal Logics and Fibring Neural Networks
- 构建兼容神经网络融合的纤维化模型，实现符号逻辑与神经网络的统一形式框架。
- 证明图神经网络、注意力机制和变换器在逻辑表达上的非均匀能力差异。
- 为理解神经网络学习到的逻辑理论提供计算逻辑分析工具，适合逻辑与机器学习交叉研究者。
模态逻辑的纤维化是一种成熟的形式化方法，可将可数个模态逻辑组合成具有共同语义的单一纤维语言，由纤维化模型表征。受此启发，神经网络的纤维化被提出作为结合学习与推理的神经符号框架。该方法利用训练后网络的（预）激活值，通过纤维化函数计算另一网络的权重，并将输出反注入原网络。然而，神经网络纤维化与模态逻辑纤维化之间的精确对应关系始终未被正式确立。本文通过形式化定义与纤维化神经网络相容的纤维化模型，填补了这一空白。基于此对应关系，我们推导出图神经网络（GNN）、图注意力网络（GAT）及变换器编码器的非均匀逻辑表达能力结果。长期目标是将纤维化作为形式化工具，用于解析神经网络所学习到的逻辑理论，借助计算逻辑的分析手段进行解释。
原文摘要 · Abstract (English)
Fibring of modal logics is a well-established formalism for combining countable families of modal logics into a single fibred language with common semantics, characterized by fibred models. Inspired by this formalism, fibring of neural networks was introduced as a neurosymbolic framework for combining learning and reasoning in neural networks. Fibring of neural networks uses the (pre-)activations of a trained network to evaluate a fibring function computing the weights of another network whose outputs are injected back into the original network. However, the exact correspondence between fibring of neural networks and fibring of modal logics was never formally established. In this paper, we close this gap by formalizing the idea of fibred models \emph{compatible} with fibred neural networks. Using this correspondence, we then derive non-uniform logical expressiveness results for Graph Neural Networks (GNNs), Graph Attention Networks (GATs) and Transformer encoders. Longer-term, the goal of this paper is to open the way for the use of fibring as a formalism for interpreting the logical theories learnt by neural networks with the tools of computational logic.
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