L{}ukasiewicz Neural Networks Extended: Residual Architectures and Crystallization Strategies for Interpretable Rule Extraction

arXiv:2609.33028v1 Announce Type: new Abstract: A feed-forward neural network whose weights are integers and whose activation is the truncated identity implements, neuron by neuron, the connectives of L{}ukasiewicz 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 L{}ukasiewicz neural networks to arbitrary depth while preserving the symbolic correspondence at merge neurons by construction: merge neurons in a L{}ukasiewicz 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.

Source: arXiv cs.LG | 2026-09-29