The AI community is increasingly focused on merging logic with deep learning to create Neuro-Symbolic (NeSy) paradigms and assist neural approaches with symbolic knowledge. A significant trend in the literature involves integrating axioms and facts in loss functions by grounding logical symbols with neural networks and operators with fuzzy semantics. Logic Tensor Networks (LTN) is one of the main representatives in this category, known for its simplicity, efficiency, and versatility. However, it has been previously shown that not all fuzzy operators perform equally when applied in a differentiable setting. Researchers have proposed several configurations of operators, trading off between effectiveness, numerical stability, and generalization to different formulas. This paper presents a configuration of fuzzy operators for grounding formulas end-to-end in the logarithm space. Our goal is to develop a configuration that is more effective than previous proposals, able to handle any formula, and numerically stable. To achieve this, we propose semantics that are best suited for the logarithm space and introduce novel simplifications and improvements that are crucial for optimization via gradient-descent. We use LTN as the framework for our experiments, but the conclusions of our work apply to any similar NeSy framework. Our findings, both formal and empirical, show that the proposed configuration outperforms the state-of-the-art and that each of our modifications is essential in achieving these results.

翻译：人工智能界日益关注将逻辑与深度学习相融合，以创建神经符号（NeSy）范式并借助符号知识辅助神经网络方法。文献中的一个重要趋势是通过神经网络对逻辑符号进行实例化、利用模糊语义对算子进行实例化，从而将公理和事实整合到损失函数中。逻辑张量网络（LTN）是该范畴的主要代表之一，以其简洁性、高效性和通用性而著称。然而，已有研究表明，并非所有模糊算子在可微设定下均能表现一致。研究人员提出了多种算子配置，在有效性、数值稳定性及对不同公式的泛化能力之间进行权衡。本文提出了一种在对数空间中进行端到端公式实例化的模糊算子配置。我们的目标是开发一种比先前方案更有效、能处理任意公式且数值稳定的配置。为此，我们提出了最适用于对数空间的语义，并引入了对梯度下降优化至关重要的新颖简化与改进。我们采用LTN作为实验框架，但本文结论适用于任何类似的NeSy框架。我们的形式化与实证结果表明，所提出的配置优于现有最优方法，且每一项修改对取得这些成果均不可或缺。