# Logical Neural Networks

arXiv: https://arxiv.org/abs/2006.13155 · HTML: https://mlsanity.com/paper/2006.13155/

## Key facts

- Cited by 119 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 30 of 119)

- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey](https://mlsanity.com/paper/2108.11451/) (2021)
- [Incorporating Domain Knowledge into Deep Neural Networks](https://mlsanity.com/paper/2103.00180/) (2021)
- [Interpretable and Explainable Logical Policies via Neurally Guided Symbolic Abstraction](https://mlsanity.com/paper/2306.01439/) (2023)
- [Is Neuro-Symbolic AI Meeting its Promise in Natural Language Processing? A Structured Review](https://mlsanity.com/paper/2202.12205/) (2022)
- [Neuro-Symbolic Inductive Logic Programming with Logical Neural Networks](https://mlsanity.com/paper/2112.03324/) (2021)
- [Neuro-Symbolic Reinforcement Learning with First-Order Logic](https://mlsanity.com/paper/2110.10963/) (2021)
- [Differentiable Inductive Logic Programming for Structured Examples](https://mlsanity.com/paper/2103.01719/) (2021)
- [Towards Cognitive AI Systems: a Survey and Prospective on Neuro-Symbolic AI](https://mlsanity.com/paper/2401.01040/) (2024)
- [Deep Symbolic Learning: Discovering Symbols and Rules from Perceptions](https://mlsanity.com/paper/2208.11561/) (2022)
- [Neuro-Symbolic Causal Reasoning Meets Signaling Game for Emergent Semantic Communications](https://mlsanity.com/paper/2210.12040/) (2022)
- [Neurosymbolic AI for Reasoning over Knowledge Graphs: A Survey](https://mlsanity.com/paper/2302.07200/) (2023)
- [Logic Embeddings for Complex Query Answering](https://mlsanity.com/paper/2103.00418/) (2021)
- [A Universal Question-Answering Platform for Knowledge Graphs](https://mlsanity.com/paper/2303.00595/) (2023)
- [Reasoning in Computer Vision: Taxonomy, Models, Tasks, and Methodologies](https://mlsanity.com/paper/2508.10523/) (2025)
- [Neuro-Symbolic Forward Reasoning](https://mlsanity.com/paper/2110.09383/) (2021)
- [Joint Edge-Model Sparse Learning is Provably Efficient for Graph Neural Networks](https://mlsanity.com/paper/2302.02922/) (2023)
- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [FF-NSL: Feed-Forward Neural-Symbolic Learner](https://mlsanity.com/paper/2106.13103/) (2021)
- [Neuro-Symbolic Learning of Answer Set Programs from Raw Data](https://mlsanity.com/paper/2205.12735/) (2022)
- [Reliable Natural Language Understanding with Large Language Models and Answer Set Programming](https://mlsanity.com/paper/2302.03780/) (2023)
- [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)
- [LNN-EL: A Neuro-Symbolic Approach to Short-text Entity Linking](https://mlsanity.com/paper/2106.09795/) (2021)
- [Knowledge-augmented Deep Learning and Its Applications: A Survey](https://mlsanity.com/paper/2212.00017/) (2022)
- [Learning Interpretable Differentiable Logic Networks](https://mlsanity.com/paper/2407.04168/) (2024)
- [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)
- [pix2rule: End-to-end Neuro-symbolic Rule Learning](https://mlsanity.com/paper/2106.07487/) (2021)
- [Neuro-symbolic Models for Interpretable Time Series Classification using Temporal Logic Description](https://mlsanity.com/paper/2209.09114/) (2022)

## Abstract (arXiv)

We propose a novel framework seamlessly providing key properties of both neural nets (learning) and symbolic logic (knowledge and reasoning). Every neuron has a meaning as a component of a formula in a weighted real-valued logic, yielding a highly intepretable disentangled representation. Inference is omnidirectional rather than focused on predefined target variables, and corresponds to logical reasoning, including classical first-order logic theorem proving as a special case. The model is end-to-end differentiable, and learning minimizes a novel loss function capturing logical contradiction, yielding resilience to inconsistent knowledge. It also enables the open-world assumption by maintaining bounds on truth values which can have probabilistic semantics, yielding resilience to incomplete knowledge.