# Learning Symmetric Rules with SATNet

arXiv: https://arxiv.org/abs/2206.13998 · HTML: https://mlsanity.com/paper/2206.13998/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 28 Jun 2022.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [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)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Techniques for Symbol Grounding with SATNet](https://mlsanity.com/paper/2106.11072/) (2021)
- [The Mixing method: low-rank coordinate descent for semidefinite programming with diagonal constraints](https://mlsanity.com/paper/1706.00476/) (2017)
- [DeepLogic: Towards End-to-End Differentiable Logical Reasoning](https://mlsanity.com/paper/1805.07433/) (2018)

## Built on by (most cited first, 4 of 4)

- [Neuro-symbolic Learning Yielding Logical Constraints](https://mlsanity.com/paper/2410.20957/) (2024)
- [Scalable Coupling of Deep Learning with Logical Reasoning](https://mlsanity.com/paper/2305.07617/) (2023)
- [Learning Reliable Logical Rules with SATNet](https://mlsanity.com/paper/2310.02133/) (2023)
- [Scaling Neuro-symbolic Problem Solving: Solver-Free Learning of Constraints and Objectives](https://mlsanity.com/paper/2508.20978/) (2025)

## Abstract (arXiv)

SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, just from input and output pairs where inputs are given as images. In this paper, we show how to improve the learning of SATNet by exploiting symmetries in the target rules of a given but unknown logical puzzle or more generally a logical formula. We present SymSATNet, a variant of SATNet that translates the given symmetries of the target rules to a condition on the parameters of SATNet and requires that the parameters should have a particular parametric form that guarantees the condition. The requirement dramatically reduces the number of parameters to learn for the rules with enough symmetries, and makes the parameter learning of SymSATNet much easier than that of SATNet. We also describe a technique for automatically discovering symmetries of the target rules from examples. Our experiments with Sudoku and Rubik's cube show the substantial improvement of SymSATNet over the baseline SATNet.