# Techniques for Symbol Grounding with SATNet

arXiv: https://arxiv.org/abs/2106.11072 · HTML: https://mlsanity.com/paper/2106.11072/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 49 references that are in the map.
- First posted on arXiv 16 Jun 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Do Deep Nets Really Need to be Deep?](https://mlsanity.com/paper/1312.6184/) (2013)
- [On Evaluating Adversarial Robustness](https://mlsanity.com/paper/1902.06705/) (2019)
- [Policy Distillation](https://mlsanity.com/paper/1511.06295/) (2015)
- [A Survey on Explainable Artificial Intelligence (XAI): Towards Medical XAI](https://mlsanity.com/paper/1907.07374/) (2019)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Neurosymbolic AI: The 3rd Wave](https://mlsanity.com/paper/2012.05876/) (2020)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [Learning a SAT Solver from Single-Bit Supervision](https://mlsanity.com/paper/1802.03685/) (2018)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Graph Neural Networks Meet Neural-Symbolic Computing: A Survey and Perspective](https://mlsanity.com/paper/2003.00330/) (2020)
- [Failures of Gradient-Based Deep Learning](https://mlsanity.com/paper/1703.07950/) (2017)
- [Recurrent Relational Networks](https://mlsanity.com/paper/1711.08028/) (2017)
- [Opportunities and Challenges in Deep Learning Adversarial Robustness: A Survey](https://mlsanity.com/paper/2007.00753/) (2020)
- [A Survey on Self-supervised Pre-training for Sequential Transfer Learning in Neural Networks](https://mlsanity.com/paper/2007.00800/) (2020)
- [Tunneling Neural Perception and Logic Reasoning through Abductive Learning](https://mlsanity.com/paper/1802.01173/) (2018)
- [An Integer Linear Programming Framework for Mining Constraints from Data](https://mlsanity.com/paper/2006.10836/) (2020)

## Built on by (most cited first, 24 of 24)

- [Deep Symbolic Learning: Discovering Symbols and Rules from Perceptions](https://mlsanity.com/paper/2208.11561/) (2022)
- [Learning to Solve Constraint Satisfaction Problems with Recurrent Transformer](https://mlsanity.com/paper/2307.04895/) (2023)
- [Softened Symbol Grounding for Neuro-symbolic Systems](https://mlsanity.com/paper/2403.00323/) (2024)
- [Neural Concept Binder](https://mlsanity.com/paper/2406.09949/) (2024)
- [From Perception to Programs: Regularize, Overparameterize, and Amortize](https://mlsanity.com/paper/2206.05922/) (2022)
- [Neuro-symbolic Learning Yielding Logical Constraints](https://mlsanity.com/paper/2410.20957/) (2024)
- [Probabilistic Abduction for Visual Abstract Reasoning via Learning Rules in Vector-symbolic Architectures](https://mlsanity.com/paper/2401.16024/) (2024)
- [Learning Guided Automated Reasoning: A Brief Survey](https://mlsanity.com/paper/2403.04017/) (2024)
- [Learning Symmetric Rules with SATNet](https://mlsanity.com/paper/2206.13998/) (2022)
- [Learning Reliable Logical Rules with SATNet](https://mlsanity.com/paper/2310.02133/) (2023)
- [Teaching Transformers to Solve Combinatorial Problems through Efficient Trial & Error](https://mlsanity.com/paper/2509.22023/) (2025)
- [Generating by Understanding: Neural Visual Generation with Logical Symbol Groundings](https://mlsanity.com/paper/2310.17451/) (2023)
- [Simple and Effective Transfer Learning for Neuro-Symbolic Integration](https://mlsanity.com/paper/2402.14047/) (2024)
- [Inductive Learning of Robot Task Knowledge from Raw Data and Online Expert Feedback](https://mlsanity.com/paper/2501.07507/) (2025)
- [Differentiable Rule Induction from Raw Sequence Inputs](https://mlsanity.com/paper/2602.13583/) (2026)
- [Differentiable Fuzzy ALC: A Neural-Symbolic Representation Language for Symbol Grounding](https://mlsanity.com/paper/2211.12006/) (2022)
- [Symbol Correctness in Deep Neural Networks Containing Symbolic Layers](https://mlsanity.com/paper/2402.03663/) (2024)
- [Logic of Hypotheses: from Zero to Full Knowledge in Neurosymbolic Integration](https://mlsanity.com/paper/2509.21663/) (2025)
- [Learning Modulo Theories](https://mlsanity.com/paper/2301.11435/) (2023)
- [A Recursive Bateson-Inspired Model for the Generation of Semantic Formal Concepts from Spatial Sensory Data](https://mlsanity.com/paper/2307.08087/) (2023)
- [Scaling Neuro-symbolic Problem Solving: Solver-Free Learning of Constraints and Objectives](https://mlsanity.com/paper/2508.20978/) (2025)
- [Visual Perceptual to Conceptual First-Order Rule Learning Networks](https://mlsanity.com/paper/2604.07897/) (2026)
- [Grounding vs. Compositionality: On the Non-Complementarity of Reasoning in Neuro-Symbolic Systems](https://mlsanity.com/paper/2604.26521/) (2026)
- [Position: Certified Correctness in Neural Constraint Reasoning Requires Symbolic Integration](https://mlsanity.com/paper/2608.14569/) (2026)

## Abstract (arXiv)

Many experts argue that the future of artificial intelligence is limited by the field's ability to integrate symbolic logical reasoning into deep learning architectures. The recently proposed differentiable MAXSAT solver, SATNet, was a breakthrough in its capacity to integrate with a traditional neural network and solve visual reasoning problems. For instance, it can learn the rules of Sudoku purely from image examples. Despite its success, SATNet was shown to succumb to a key challenge in neurosymbolic systems known as the Symbol Grounding Problem: the inability to map visual inputs to symbolic variables without explicit supervision ("label leakage"). In this work, we present a self-supervised pre-training pipeline that enables SATNet to overcome this limitation, thus broadening the class of problems that SATNet architectures can solve to include datasets where no intermediary labels are available at all. We demonstrate that our method allows SATNet to attain full accuracy even with a harder problem setup that prevents any label leakage. We additionally introduce a proofreading method that further improves the performance of SATNet architectures, beating the state-of-the-art on Visual Sudoku.