# logLTN: Differentiable Fuzzy Logic in the Logarithm Space

arXiv: https://arxiv.org/abs/2306.14546 · HTML: https://mlsanity.com/paper/2306.14546/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 26 Jun 2023.

## Builds on (most cited first, 14 of 14)

- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Neurosymbolic AI: The 3rd Wave](https://mlsanity.com/paper/2012.05876/) (2020)
- [Detect What You Can: Detecting and Representing Objects using Holistic Models and Body Parts](https://mlsanity.com/paper/1406.2031/) (2014)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Analyzing Differentiable Fuzzy Logic Operators](https://mlsanity.com/paper/2002.06100/) (2020)
- [Complex Query Answering with Neural Link Predictors](https://mlsanity.com/paper/2011.03459/) (2020)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [Neuro-Symbolic Artificial Intelligence: Current Trends](https://mlsanity.com/paper/2105.05330/) (2021)
- [Fuzzy Logic Based Logical Query Answering on Knowledge Graphs](https://mlsanity.com/paper/2108.02390/) (2021)
- [Faster-LTN: a neuro-symbolic, end-to-end object detection architecture](https://mlsanity.com/paper/2107.01877/) (2021)
- [Semi-Supervised Learning using Differentiable Reasoning](https://mlsanity.com/paper/1908.04700/) (2019)
- [Injecting Prior Knowledge for Transfer Learning into Reinforcement Learning Algorithms using Logic Tensor Networks](https://mlsanity.com/paper/1906.06576/) (2019)

## Built on by (most cited first, 5 of 5)

- [Zero-Shot Conditioning of Score-Based Diffusion Models by Neuro-Symbolic Constraints](https://mlsanity.com/paper/2308.16534/) (2023)
- [Differentiable Constraint-Based Causal Discovery](https://mlsanity.com/paper/2510.22031/) (2025)
- [T-Norm Operators for EU AI Act Compliance Classification: An Empirical Comparison of Lukasiewicz, Product, and G"odel Semantics in a Neuro-Symbolic Reasoning System](https://mlsanity.com/paper/2603.28558/) (2026)
- [NeSyCat Torch: A Differentiable Tensor Implementation of Categorical Semantics for Neurosymbolic Learning](https://mlsanity.com/paper/2606.19279/) (2026)
- [Neurosymbolic Learning of Interpretable World Models](https://mlsanity.com/paper/2610.19955/) (2026)

## Abstract (arXiv)

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.