发表机构
The University of Edinburgh（爱丁堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对LTN-GAN的硬约束问题，提出将公理以函数符号而非谓词 grounding 的方法，对比约束层发现其能保留余量分布，通过分辨率比R诊断 grounding 的学习能力，可生成真实有效样本。
AI 中文摘要
逻辑张量网络增强生成对抗网络（LTN-GAN）通过将每个逻辑公理 grounding 为谓词并训练生成器提高其满意度（即[0,1]区间内的模糊真值）来注入背景知识。此前的LTN-GAN工作在谓词层面以这种方式 grounding 所有约束，提升了约束满意度。然而，谓词仅能对样本打分，无法嵌入硬结构约束，即每个生成样本都必须满足的规则，如有序性、正性和定义恒等式。本研究探讨在LTN框架内将每个公理 grounding 为函数符号。我们将其与最先进的替代方案——约束层进行对比，该约束层会将每个违反约束的样本钳制到可行边界，从而产生始终有效的输出。研究表明，有效样本未必是真实样本；不等式并非仅满足或违反，而是存在一定余量，忠实的生成器应复现余量的真实分布。我们发现分辨率比R（数据尺度与余量分布范围之比）是一种诊断指标，可在训练前计算，用于判断所选 grounding 能学习哪些约束。当R较大时，谓词无法获取学习信号，钳制操作会将所有样本推到边界，导致余量分布丢失，而所有标准指标仍表现良好。函数符号可避免这两种失败，它会计算受约束变量而非对其打分。函数符号共同构成一个图表，即可行区域内的坐标系，其中每个样本天生有效，且余量会像其他量一样被学习。
英文摘要
Logic Tensor Network-Enhanced Generative Adversarial Networks (LTN-GANs) inject background knowledge by grounding each logical axiom as a predicate and training the generator to raise its satisfaction, a fuzzy truth value in $[0,1]$. Previous LTN-GAN work grounded every constraint this way, at the predicate level, and improved constraint satisfaction. A predicate, however, only scores a sample, so it cannot embed hard structural constraints, rules such as orderings, positivity, and definitional identities that must hold in every generated sample. In this work, we investigate grounding each axiom as a function symbol inside the LTN framework. We compare against the state-of-the-art alternative, a constraint layer that clamps each violating sample onto the feasible boundary and so produces outputs that are always valid. Our investigation shows that a valid sample is not always a realistic one. An inequality is not merely satisfied or violated. It holds by a margin, and a faithful generator should also reproduce the margin's real distribution. We find that the resolution ratio $R$, the data's scale over the margin's spread, is a diagnostic, computable before training, of which constraints a chosen grounding can learn. When $R$ is large, the predicate receives no learning signal, the clamp pushes every sample onto the boundary, and the margin distribution is lost while every standard metric still looks fine. A function symbol avoids both failures, computing the constrained variable rather than scoring it. Together the function symbols form a chart, a coordinate system inside the feasible region, where every sample is valid by construction and the margin is learned like any other quantity.
