# FF-NSL: Feed-Forward Neural-Symbolic Learner

arXiv: https://arxiv.org/abs/2106.13103 · HTML: https://mlsanity.com/paper/2106.13103/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 78 references that are in the map.
- First posted on arXiv 24 Jun 2021.

## Builds on (most cited first, 26 of 26)

- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Explaining Explanations: An Overview of Interpretability of Machine Learning](https://mlsanity.com/paper/1806.00069/) (2018)
- [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)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [Neural-Symbolic Learning and Reasoning: A Survey and Interpretation](https://mlsanity.com/paper/1711.03902/) (2017)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [An open access repository of images on plant health to enable the development of mobile disease diagnostics](https://mlsanity.com/paper/1511.08060/) (2015)
- [Logical Neural Networks](https://mlsanity.com/paper/2006.13155/) (2020)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [Understanding Softmax Confidence and Uncertainty](https://mlsanity.com/paper/2106.04972/) (2021)
- [Learning Reasoning Strategies in End-to-End Differentiable Proving](https://mlsanity.com/paper/2007.06477/) (2020)
- [Neural-Symbolic Integration: A Compositional Perspective](https://mlsanity.com/paper/2010.11926/) (2020)
- [Semantic-Aware Scene Recognition](https://mlsanity.com/paper/1909.02410/) (2019)
- [Turning 30: New Ideas in Inductive Logic Programming](https://mlsanity.com/paper/2002.11002/) (2020)
- [Uncertainty-Aware Deep Classifiers using Generative Models](https://mlsanity.com/paper/2006.04183/) (2020)
- [Abductive Knowledge Induction From Raw Data](https://mlsanity.com/paper/2010.03514/) (2020)
- [Differentiable Reasoning on Large Knowledge Bases and Natural Language](https://mlsanity.com/paper/1912.10824/) (2019)
- [TensorLog: A Differentiable Deductive Database](https://mlsanity.com/paper/1605.06523/) (2016)
- [Inductive Learning of Answer Set Programs from Noisy Examples](https://mlsanity.com/paper/1808.08441/) (2018)
- [Structure Learning of Probabilistic Logic Programs by Searching the Clause Space](https://mlsanity.com/paper/1309.2080/) (2013)

## Built on by (most cited first, 15 of 15)

- [Neuro-Symbolic Learning of Answer Set Programs from Raw Data](https://mlsanity.com/paper/2205.12735/) (2022)
- [Learning Differentiable Logic Programs for Abstract Visual Reasoning](https://mlsanity.com/paper/2307.00928/) (2023)
- [The Role of Foundation Models in Neuro-Symbolic Learning and Reasoning](https://mlsanity.com/paper/2402.01889/) (2024)
- [Neuro-Symbolic AI: Explainability, Challenges, and Future Trends](https://mlsanity.com/paper/2411.04383/) (2024)
- [V-LoL: A Diagnostic Dataset for Visual Logical Learning](https://mlsanity.com/paper/2306.07743/) (2023)
- [Learning Robust Reward Machines from Noisy Labels](https://mlsanity.com/paper/2408.14871/) (2024)
- [Neural Meta-Symbolic Reasoning and Learning](https://mlsanity.com/paper/2211.11650/) (2022)
- [Abductive Logical Rule Induction by Bridging Inductive Logic Programming and Multimodal Large Language Models](https://mlsanity.com/paper/2509.21874/) (2025)
- [Differentiable Inductive Logic Programming in High-Dimensional Space](https://mlsanity.com/paper/2208.06652/) (2022)
- [Neuro-Symbolic Contrastive Learning for Cross-domain Inference](https://mlsanity.com/paper/2502.09213/) (2025)
- [Disentangling Neural Disjunctive Normal Form Models](https://mlsanity.com/paper/2507.10546/) (2025)
- [LearnAD: Learning Interpretable Rules for Brain Networks in Alzheimer's Disease Classification](https://mlsanity.com/paper/2601.00877/) (2025)
- [Accelerating NeurASP with vectorization and caching](https://mlsanity.com/paper/2606.10787/) (2026)
- [LLM and Human Modes of Representation](https://mlsanity.com/paper/2606.21616/) (2026)
- [Pretrain on Small Synthetic Data, Scale Large for Free: Symmetry-Aware Foundation Model for Logic Rule Induction](https://mlsanity.com/paper/2608.00383/) (2026)

## Abstract (arXiv)

Logic-based machine learning aims to learn general, interpretable knowledge in a data-efficient manner. However, labelled data must be specified in a structured logical form. To address this limitation, we propose a neural-symbolic learning framework, called Feed-Forward Neural-Symbolic Learner (FFNSL), that integrates a logic-based machine learning system capable of learning from noisy examples, with neural networks, in order to learn interpretable knowledge from labelled unstructured data. We demonstrate the generality of FFNSL on four neural-symbolic classification problems, where different pre-trained neural network models and logic-based machine learning systems are integrated to learn interpretable knowledge from sequences of images. We evaluate the robustness of our framework by using images subject to distributional shifts, for which the pre-trained neural networks may predict incorrectly and with high confidence. We analyse the impact that these shifts have on the accuracy of the learned knowledge and run-time performance, comparing FFNSL to tree-based and pure neural approaches. Our experimental results show that FFNSL outperforms the baselines by learning more accurate and interpretable knowledge with fewer examples.