# Lukasiewicz Neural Networks Extended: Residual Architectures and Crystallization Strategies for Interpretable Rule Extraction

arXiv: https://arxiv.org/abs/2609.33028 · HTML: https://mlsanity.com/paper/2609.33028/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 26 Sep 2026.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [A Survey Of Methods For Explaining Black Box Models](https://mlsanity.com/paper/1802.01933/) (2018)
- [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)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [Ternary Weight Networks](https://mlsanity.com/paper/1605.04711/) (2016)
- [Neural-Symbolic Computing: An Effective Methodology for Principled Integration of Machine Learning and Reasoning](https://mlsanity.com/paper/1905.06088/) (2019)
- [Logical Neural Networks](https://mlsanity.com/paper/2006.13155/) (2020)
- [Symbolic Knowledge Extraction using Lukasiewicz Logics](https://mlsanity.com/paper/1604.03099/) (2016)

## Abstract (arXiv)

A feed-forward neural network whose weights are integers and whose activation is the truncated identity implements, neuron by neuron, the connectives of Lukasiewicz many-valued logic. This exact correspondence --- established theoretically by Castro and Trillas and developed into a training algorithm by Leandro --- enables symbolic knowledge extraction: training produces not a black-box model but a logical formula. Two obstacles have limited the approach to shallow architectures and small datasets: crystallization (forcing weights to integers) succeeds only probabilistically under the original Levenberg--Marquardt training scheme, and the theoretical guarantees break down as networks grow deeper. This paper addresses both obstacles. First, we prove that residual connections (skip connections of the kind used in ResNets) extend Lukasiewicz neural networks to arbitrary depth while preserving the symbolic correspondence at merge neurons by construction: merge neurons in a Lukasiewicz residual block automatically satisfy the neuron-classification proposition, regardless of the inner layer weights; inner-layer neurons are trained toward representability by the crystallization strategy. Second, we analyse three crystallization strategies --- Levenberg--Marquardt (corrected), straight-through estimation (STE), and proximal regularization --- characterizing their theoretical guarantees, failure modes, and interpretability trade-offs.