# Neuro-symbolic Learning Yielding Logical Constraints

arXiv: https://arxiv.org/abs/2410.20957 · HTML: https://mlsanity.com/paper/2410.20957/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 28 of its 76 references that are in the map.
- First posted on arXiv 28 Oct 2024.

## Placed in

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

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [nuScenes: A multimodal dataset for autonomous driving](https://mlsanity.com/paper/1903.11027/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [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)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Neural Logic Machines](https://mlsanity.com/paper/1904.11694/) (2019)
- [Inductive logic programming at 30: a new introduction](https://mlsanity.com/paper/2008.07912/) (2020)
- [Failures of Gradient-Based Deep Learning](https://mlsanity.com/paper/1703.07950/) (2017)
- [Recurrent Relational Networks](https://mlsanity.com/paper/1711.08028/) (2017)
- [Thinking Fast and Slow in AI](https://mlsanity.com/paper/2010.06002/) (2020)
- [The Implicit Regularization of Stochastic Gradient Flow for Least Squares](https://mlsanity.com/paper/2003.07802/) (2020)
- [Closed Loop Neural-Symbolic Learning via Integrating Neural Perception, Grammar Parsing, and Symbolic Reasoning](https://mlsanity.com/paper/2006.06649/) (2020)
- [A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference](https://mlsanity.com/paper/2212.12393/) (2022)
- [Modeling Recurrence for Transformer](https://mlsanity.com/paper/1904.03092/) (2019)
- [Techniques for Symbol Grounding with SATNet](https://mlsanity.com/paper/2106.11072/) (2021)
- [Learning to Solve Constraint Satisfaction Problems with Recurrent Transformer](https://mlsanity.com/paper/2307.04895/) (2023)
- [Learning a Decision Module by Imitating Driver's Control Behaviors](https://mlsanity.com/paper/1912.00191/) (2019)
- [Softened Symbol Grounding for Neuro-symbolic Systems](https://mlsanity.com/paper/2403.00323/) (2024)
- [Low-rank semidefinite programming for the MAX2SAT problem](https://mlsanity.com/paper/1812.06362/) (2018)
- [Exact Semidefinite Formulations for a Class of (Random and Non-Random) Nonconvex Quadratic Programs](https://mlsanity.com/paper/1802.02688/) (2018)
- [On the tightness of SDP relaxations of QCQPs](https://mlsanity.com/paper/1911.09195/) (2019)
- [Learning Symmetric Rules with SATNet](https://mlsanity.com/paper/2206.13998/) (2022)
- [Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas](https://mlsanity.com/paper/1711.01358/) (2017)

## Built on by (most cited first, 12 of 12)

- [LogiCity: Advancing Neuro-Symbolic AI with Abstract Urban Simulation](https://mlsanity.com/paper/2411.00773/) (2024)
- [Towards Trustworthy Legal AI through LLM Agents and Formal Reasoning](https://mlsanity.com/paper/2511.21033/) (2025)
- [A Learnability Analysis on Neuro-Symbolic Learning](https://mlsanity.com/paper/2503.16797/) (2025)
- [Logic Tensor Network-Enhanced Generative Adversarial Network](https://mlsanity.com/paper/2601.03839/) (2026)
- [Neurosymbolic LoRA: Why and When to Tune Weights vs. Rewrite Prompts](https://mlsanity.com/paper/2601.12711/) (2026)
- [Process-of-Thought Reasoning for Videos](https://mlsanity.com/paper/2602.07689/) (2026)
- [Interactionless Inverse Reinforcement Learning: A Data-Centric Framework for Durable Alignment](https://mlsanity.com/paper/2602.14844/) (2026)
- [Solving Satisfiability Modulo Counting Exactly with Probabilistic Circuits](https://mlsanity.com/paper/2503.01009/) (2025)
- [Constraints-Guided Diffusion Reasoner for Neuro-Symbolic Learning](https://mlsanity.com/paper/2508.16524/) (2025)
- [Scaling Neuro-symbolic Problem Solving: Solver-Free Learning of Constraints and Objectives](https://mlsanity.com/paper/2508.20978/) (2025)
- [Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs](https://mlsanity.com/paper/2608.21605/) (2026)
- [Prototype-Rule Neurosymbolic Regularization for Rank-Constrained Tensor Neural Networks under Label Scarcity](https://mlsanity.com/paper/2609.40131/) (2026)

## Abstract (arXiv)

Neuro-symbolic systems combine the abilities of neural perception and logical reasoning. However, end-to-end learning of neuro-symbolic systems is still an unsolved challenge. This paper proposes a natural framework that fuses neural network training, symbol grounding, and logical constraint synthesis into a coherent and efficient end-to-end learning process. The capability of this framework comes from the improved interactions between the neural and the symbolic parts of the system in both the training and inference stages. Technically, to bridge the gap between the continuous neural network and the discrete logical constraint, we introduce a difference-of-convex programming technique to relax the logical constraints while maintaining their precision. We also employ cardinality constraints as the language for logical constraint learning and incorporate a trust region method to avoid the degeneracy of logical constraint in learning. Both theoretical analyses and empirical evaluations substantiate the effectiveness of the proposed framework.