logLTN: Differentiable Fuzzy Logic in the Logarithm Space

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.