# Injecting Logical Constraints into Neural Networks via Straight-Through Estimators

arXiv: https://arxiv.org/abs/2307.04347 · HTML: https://mlsanity.com/paper/2307.04347/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 38 references that are in the map.
- First posted on arXiv 10 Jul 2023.

## Builds on (most cited first, 18 of 18)

- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [BinaryConnect: Training Deep Neural Networks with binary weights during propagations](https://mlsanity.com/paper/1511.00363/) (2015)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [The Neuro-Symbolic Concept Learner: Interpreting Scenes, Words, and Sentences From Natural Supervision](https://mlsanity.com/paper/1904.12584/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Learning a SAT Solver from Single-Bit Supervision](https://mlsanity.com/paper/1802.03685/) (2018)
- [Neural-Symbolic Learning and Reasoning: A Survey and Interpretation](https://mlsanity.com/paper/1711.03902/) (2017)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Neural-Symbolic Computing: An Effective Methodology for Principled Integration of Machine Learning and Reasoning](https://mlsanity.com/paper/1905.06088/) (2019)
- [Complex Query Answering with Neural Link Predictors](https://mlsanity.com/paper/2011.03459/) (2020)
- [Graph Neural Networks Meet Neural-Symbolic Computing: A Survey and Perspective](https://mlsanity.com/paper/2003.00330/) (2020)
- [Recurrent Relational Networks](https://mlsanity.com/paper/1711.08028/) (2017)
- [From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey](https://mlsanity.com/paper/2108.11451/) (2021)
- [Neural-Symbolic Integration: A Compositional Perspective](https://mlsanity.com/paper/2010.11926/) (2020)
- [Plug-in, Trainable Gate for Streamlining Arbitrary Neural Networks](https://mlsanity.com/paper/1904.10921/) (2019)

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

- [Learning with Logical Constraints but without Shortcut Satisfaction](https://mlsanity.com/paper/2403.00329/) (2024)
- [Efficient Rectification of Neuro-Symbolic Reasoning Inconsistencies by Abductive Reflection](https://mlsanity.com/paper/2412.08457/) (2024)
- [Self-Supervised Transformers as Iterative Solution Improvers for Constraint Satisfaction](https://mlsanity.com/paper/2502.15794/) (2025)
- [AlignSAE: Concept-Aligned Sparse Autoencoders](https://mlsanity.com/paper/2512.02004/) (2025)
- [LogicMP: A Neuro-symbolic Approach for Encoding First-order Logic Constraints](https://mlsanity.com/paper/2309.15458/) (2023)
- [Logic Tensor Network-Enhanced Generative Adversarial Network](https://mlsanity.com/paper/2601.03839/) (2026)
- [Edge Detectors Can Make Deep Convolutional Neural Networks More Robust](https://mlsanity.com/paper/2402.16479/) (2024)
- [Beyond Discreteness: Sample Complexity Analysis of Straight-Through Estimator for 1-bit Quantization](https://mlsanity.com/paper/2505.18113/) (2025)
- [Learning-driven Physically-aware Large-scale Circuit Gate Sizing](https://mlsanity.com/paper/2403.08193/) (2024)
- [Curriculum Abductive Learning](https://mlsanity.com/paper/2505.12275/) (2025)
- [Neurosymbolic LoRA: Why and When to Tune Weights vs. Rewrite Prompts](https://mlsanity.com/paper/2601.12711/) (2026)
- [Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs](https://mlsanity.com/paper/2608.21605/) (2026)

## Abstract (arXiv)

Injecting discrete logical constraints into neural network learning is one of the main challenges in neuro-symbolic AI. We find that a straight-through-estimator, a method introduced to train binary neural networks, could effectively be applied to incorporate logical constraints into neural network learning. More specifically, we design a systematic way to represent discrete logical constraints as a loss function; minimizing this loss using gradient descent via a straight-through-estimator updates the neural network's weights in the direction that the binarized outputs satisfy the logical constraints. The experimental results show that by leveraging GPUs and batch training, this method scales significantly better than existing neuro-symbolic methods that require heavy symbolic computation for computing gradients. Also, we demonstrate that our method applies to different types of neural networks, such as MLP, CNN, and GNN, making them learn with no or fewer labeled data by learning directly from known constraints.