# Semantic Strengthening of Neuro-Symbolic Learning

arXiv: https://arxiv.org/abs/2302.14207 · HTML: https://mlsanity.com/paper/2302.14207/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 50 references that are in the map.
- First posted on arXiv 28 Feb 2023.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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 to Explain: An Information-Theoretic Perspective on Model Interpretation](https://mlsanity.com/paper/1802.07814/) (2018)
- [Explainable Prediction of Medical Codes from Clinical Text](https://mlsanity.com/paper/1802.05695/) (2018)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [Analyzing Differentiable Fuzzy Logic Operators](https://mlsanity.com/paper/2002.06100/) (2020)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [Logic-Guided Data Augmentation and Regularization for Consistent Question Answering](https://mlsanity.com/paper/2004.10157/) (2020)
- [Coherent Hierarchical Multi-Label Classification Networks](https://mlsanity.com/paper/2010.10151/) (2020)
- [Hinge-Loss Markov Random Fields and Probabilistic Soft Logic](https://mlsanity.com/paper/1505.04406/) (2015)
- [From Statistical Relational to Neuro-Symbolic Artificial Intelligence](https://mlsanity.com/paper/2003.08316/) (2020)
- [Augmenting Neural Networks with First-order Logic](https://mlsanity.com/paper/1906.06298/) (2019)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Multi-Label Classification Neural Networks with Hard Logical Constraints](https://mlsanity.com/paper/2103.13427/) (2021)
- [NeuPSL: Neural Probabilistic Soft Logic](https://mlsanity.com/paper/2205.14268/) (2022)
- [SIMPLE: A Gradient Estimator for k-Subset Sampling](https://mlsanity.com/paper/2210.01941/) (2022)
- [Programming with a Differentiable Forth Interpreter](https://mlsanity.com/paper/1605.06640/) (2016)
- [Neuro-Symbolic Entropy Regularization](https://mlsanity.com/paper/2201.11250/) (2022)
- [On Tractable Computation of Expected Predictions](https://mlsanity.com/paper/1910.02182/) (2019)
- [Adversarial Sets for Regularising Neural Link Predictors](https://mlsanity.com/paper/1707.07596/) (2017)
- [Refining neural network predictions using background knowledge](https://mlsanity.com/paper/2206.04976/) (2022)
- [Evaluating Relaxations of Logic for Neural Networks: A Comprehensive Study](https://mlsanity.com/paper/2107.13646/) (2021)
- [Tunneling Neural Perception and Logic Reasoning through Abductive Learning](https://mlsanity.com/paper/1802.01173/) (2018)
- [Conditional Mutual Information Estimation for Mixed Discrete and Continuous Variables with Nearest Neighbors](https://mlsanity.com/paper/1912.03387/) (2019)

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

- [Neuro-Symbolic AI in 2024: A Systematic Review](https://mlsanity.com/paper/2501.05435/) (2025)
- [A Pseudo-Semantic Loss for Autoregressive Models with Logical Constraints](https://mlsanity.com/paper/2312.03905/) (2023)
- [Neurosymbolic Diffusion Models](https://mlsanity.com/paper/2505.13138/) (2025)
- [Neuro-Symbolic AI: Explainability, Challenges, and Future Trends](https://mlsanity.com/paper/2411.04383/) (2024)
- [A Unified Approach to Count-Based Weakly-Supervised Learning](https://mlsanity.com/paper/2311.13718/) (2023)
- [On the Hardness of Probabilistic Neurosymbolic Learning](https://mlsanity.com/paper/2406.04472/) (2024)
- [EXPLAIN, AGREE, LEARN: Scaling Learning for Neural Probabilistic Logic](https://mlsanity.com/paper/2408.08133/) (2024)
- [PuzzlePlex: Benchmarking Foundation Models on Reasoning and Planning with Puzzles](https://mlsanity.com/paper/2510.06475/) (2025)
- [Semantic Loss Functions for Neuro-Symbolic Structured Prediction](https://mlsanity.com/paper/2405.07387/) (2024)
- [Improving Neural-based Classification with Logical Background Knowledge](https://mlsanity.com/paper/2402.13019/) (2024)
- [A Complexity Map of Probabilistic Reasoning for Neurosymbolic Classification Techniques](https://mlsanity.com/paper/2404.08404/) (2024)
- [G-RRM: Guiding Symbolic Solvers with Recurrent Reasoning Models](https://mlsanity.com/paper/2607.02491/) (2026)

## Abstract (arXiv)

Numerous neuro-symbolic approaches have recently been proposed typically with the goal of adding symbolic knowledge to the output layer of a neural network. Ideally, such losses maximize the probability that the neural network's predictions satisfy the underlying domain. Unfortunately, this type of probabilistic inference is often computationally infeasible. Neuro-symbolic approaches therefore commonly resort to fuzzy approximations of this probabilistic objective, sacrificing sound probabilistic semantics, or to sampling which is very seldom feasible. We approach the problem by first assuming the constraint decomposes conditioned on the features learned by the network. We iteratively strengthen our approximation, restoring the dependence between the constraints most responsible for degrading the quality of the approximation. This corresponds to computing the mutual information between pairs of constraints conditioned on the network's learned features, and may be construed as a measure of how well aligned the gradients of two distributions are. We show how to compute this efficiently for tractable circuits. We test our approach on three tasks: predicting a minimum-cost path in Warcraft, predicting a minimum-cost perfect matching, and solving Sudoku puzzles, observing that it improves upon the baselines while sidestepping intractability.