# Reduced Implication-bias Logic Loss for Neuro-Symbolic Learning

arXiv: https://arxiv.org/abs/2208.06838 · HTML: https://mlsanity.com/paper/2208.06838/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 14 Aug 2022.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 16 of 16)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Grounded Language-Image Pre-training](https://mlsanity.com/paper/2112.03857/) (2021)
- [When Does Label Smoothing Help?](https://mlsanity.com/paper/1906.02629/) (2019)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Neural-Symbolic Computing: An Effective Methodology for Principled Integration of Machine Learning and Reasoning](https://mlsanity.com/paper/1905.06088/) (2019)
- [Analyzing Differentiable Fuzzy Logic Operators](https://mlsanity.com/paper/2002.06100/) (2020)
- [From Statistical Relational to Neuro-Symbolic Artificial Intelligence](https://mlsanity.com/paper/2003.08316/) (2020)
- [Augmenting Neural Networks with First-order Logic](https://mlsanity.com/paper/1906.06298/) (2019)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [On the relation between Loss Functions and T-Norms](https://mlsanity.com/paper/1907.07904/) (2019)

## Built on by (most cited first, 2 of 2)

- [A Learnability Analysis on Neuro-Symbolic Learning](https://mlsanity.com/paper/2503.16797/) (2025)
- [Candidate Retention for Abductive Learning](https://mlsanity.com/paper/2609.39561/) (2026)

## Abstract (arXiv)

Integrating logical reasoning and machine learning by approximating logical inference with differentiable operators is a widely used technique in Neuro-Symbolic systems. However, some differentiable operators could bring a significant bias during backpropagation and degrade the performance of Neuro-Symbolic learning. In this paper, we reveal that this bias, named textitImplication Bias is common in loss functions derived from fuzzy logic operators. Furthermore, we propose a simple yet effective method to transform the biased loss functions into textitReduced Implication-bias Logic Loss (RILL) to address the above problem. Empirical study shows that RILL can achieve significant improvements compared with the biased logic loss functions, especially when the knowledge base is incomplete, and keeps more robust than the compared methods when labelled data is insufficient.