Łukasiewicz Neural Networks Extended: Residual Architectures and Crystallization Strategies for Interpretable Rule Extraction
Carlos Leandro
Abstract
A feed-forward neural network whose weights are integers and whose activation is the truncated identity implements, neuron by neuron, the connectives of Łukasiewicz many-valued logic. This exact correspondence --- established theoretically by Castro and Trillas and developed into a training algorithm by Leandro --- enables \emph{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 \emph{residual connections} (skip connections of the kind used in ResNets) extend Łukasiewicz neural networks to arbitrary depth while preserving the symbolic correspondence \emph{at merge neurons} by construction: merge neurons in a Łukasiewicz 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.