# Neuro-Symbolic Inductive Logic Programming with Logical Neural Networks

arXiv: https://arxiv.org/abs/2112.03324 · HTML: https://mlsanity.com/paper/2112.03324/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 35 references that are in the map.
- First posted on arXiv 6 Dec 2021.

## Builds on (most cited first, 17 of 17)

- [Convolutional 2D Knowledge Graph Embeddings](https://mlsanity.com/paper/1707.01476/) (2017)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [Maxout Networks](https://mlsanity.com/paper/1302.4389/) (2013)
- [Holographic Embeddings of Knowledge Graphs](https://mlsanity.com/paper/1510.04935/) (2015)
- [DRUM: End-To-End Differentiable Rule Mining On Knowledge Graphs](https://mlsanity.com/paper/1911.00055/) (2019)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Go for a Walk and Arrive at the Answer: Reasoning Over Paths in Knowledge Bases using Reinforcement Learning](https://mlsanity.com/paper/1711.05851/) (2017)
- [Neural Logic Machines](https://mlsanity.com/paper/1904.11694/) (2019)
- [Logical Neural Networks](https://mlsanity.com/paper/2006.13155/) (2020)
- [RNNLogic: Learning Logic Rules for Reasoning on Knowledge Graphs](https://mlsanity.com/paper/2010.04029/) (2020)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [A Re-evaluation of Knowledge Graph Completion Methods](https://mlsanity.com/paper/1911.03903/) (2019)
- [Learning Reasoning Strategies in End-to-End Differentiable Proving](https://mlsanity.com/paper/2007.06477/) (2020)
- [Differentiable Reasoning on Large Knowledge Bases and Natural Language](https://mlsanity.com/paper/1912.10824/) (2019)
- [RelNN: A Deep Neural Model for Relational Learning](https://mlsanity.com/paper/1712.02831/) (2017)
- [Stacked Structure Learning for Lifted Relational Neural Networks](https://mlsanity.com/paper/1710.02221/) (2017)

## Built on by (most cited first, 30 of 36)

- [Can LLMs Reason with Rules? Logic Scaffolding for Stress-Testing and Improving LLMs](https://mlsanity.com/paper/2402.11442/) (2024)
- [PyReason: Software for Open World Temporal Logic](https://mlsanity.com/paper/2302.13482/) (2023)
- [Interpretable Multimodal Misinformation Detection with Logic Reasoning](https://mlsanity.com/paper/2305.05964/) (2023)
- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [On Learning Latent Models with Multi-Instance Weak Supervision](https://mlsanity.com/paper/2306.13796/) (2023)
- [Neuro-Symbolic Learning of Answer Set Programs from Raw Data](https://mlsanity.com/paper/2205.12735/) (2022)
- [Neuro-Symbolic AI: Explainability, Challenges, and Future Trends](https://mlsanity.com/paper/2411.04383/) (2024)
- [DMWM: Dual-Mind World Model with Long-Term Imagination](https://mlsanity.com/paper/2502.07591/) (2025)
- [LogiCity: Advancing Neuro-Symbolic AI with Abstract Urban Simulation](https://mlsanity.com/paper/2411.00773/) (2024)
- [Neuro-Symbolic Learning: Principles and Applications in Ophthalmology](https://mlsanity.com/paper/2208.00374/) (2022)
- [Neuro-symbolic Rule Learning in Real-world Classification Tasks](https://mlsanity.com/paper/2303.16674/) (2023)
- [Reason-Align-Respond: Aligning LLM Reasoning with Knowledge Graphs for KGQA](https://mlsanity.com/paper/2505.20971/) (2025)
- [The Need for Verification in AI-Driven Scientific Discovery](https://mlsanity.com/paper/2509.01398/) (2025)
- [Behavioural vs. Representational Systematicity in End-to-End Models: An Opinionated Survey](https://mlsanity.com/paper/2506.04461/) (2025)
- [LogicCBMs: Logic-Enhanced Concept-Based Learning](https://mlsanity.com/paper/2512.07383/) (2025)
- [Learning Symbolic Rules over Abstract Meaning Representations for Textual Reinforcement Learning](https://mlsanity.com/paper/2307.02689/) (2023)
- [Using Logic Programming and Kernel-Grouping for Improving Interpretability of Convolutional Neural Networks](https://mlsanity.com/paper/2310.13073/) (2023)
- [NeSyFOLD: Neurosymbolic Framework for Interpretable Image Classification](https://mlsanity.com/paper/2301.12667/) (2023)
- [Neural Reasoning Networks: Efficient Interpretable Neural Networks With Automatic Textual Explanations](https://mlsanity.com/paper/2410.07966/) (2024)
- [Hypothesis Generation via LLM-Automated Language Bias for ILP](https://mlsanity.com/paper/2505.21486/) (2025)
- [Neural Logic Networks for Interpretable Classification](https://mlsanity.com/paper/2508.08172/) (2025)
- [Differentiable Fuzzy ALC: A Neural-Symbolic Representation Language for Symbol Grounding](https://mlsanity.com/paper/2211.12006/) (2022)
- [Three Pathways to Neurosymbolic Reinforcement Learning with Interpretable Model and Policy Networks](https://mlsanity.com/paper/2402.05307/) (2024)
- [Efficiently Learning Probabilistic Logical Models by Cheaply Ranking Mined Rules](https://mlsanity.com/paper/2409.16238/) (2024)
- [Standard Neural Computation Alone Is Insufficient for Logical Intelligence](https://mlsanity.com/paper/2502.02135/) (2025)
- [Differentiable Inductive Logic Programming in High-Dimensional Space](https://mlsanity.com/paper/2208.06652/) (2022)
- [Deep Explainable Learning with Graph Based Data Assessing and Rule Reasoning](https://mlsanity.com/paper/2211.04693/) (2022)
- [A Probabilistic-Logic based Commonsense Representation Framework for Modelling Inferences with Multiple Antecedents and Varying Likelihoods](https://mlsanity.com/paper/2211.16822/) (2022)
- [Neuro-Symbolic Contrastive Learning for Cross-domain Inference](https://mlsanity.com/paper/2502.09213/) (2025)
- [Neuro-symbolic Weak Supervision: Theory and Semantics](https://mlsanity.com/paper/2503.18509/) (2025)

## Abstract (arXiv)

Recent work on neuro-symbolic inductive logic programming has led to promising approaches that can learn explanatory rules from noisy, real-world data. While some proposals approximate logical operators with differentiable operators from fuzzy or real-valued logic that are parameter-free thus diminishing their capacity to fit the data, other approaches are only loosely based on logic making it difficult to interpret the learned "rules". In this paper, we propose learning rules with the recently proposed logical neural networks (LNN). Compared to others, LNNs offer strong connection to classical Boolean logic thus allowing for precise interpretation of learned rules while harboring parameters that can be trained with gradient-based optimization to effectively fit the data. We extend LNNs to induce rules in first-order logic. Our experiments on standard benchmarking tasks confirm that LNN rules are highly interpretable and can achieve comparable or higher accuracy due to their flexible parameterization.