Prime Convolutional Model: Breaking the Ground for Theoretical Explainability

2025-03-04

AI Summary

This work addresses the theoretical interpretability of neural network behavior. Focusing on the Prime Convolutional model (p-Conv) for recognizing modulo-$m$ congruence classes over natural number sequences, it establishes the first XAI mathematical modeling paradigm grounded in scientific methodology. The approach introduces a custom convolutional architecture with a sliding-window context of length $B$, integrates modular arithmetic task design, and employs empirically guided theoretical induction to derive a provably correct mathematical model of neural behavior. Key contributions include: (i) rigorous derivation of necessary and sufficient number-theoretic conditions—expressed in terms of $m$ and $B$—for model convergence; and (ii) analytical criteria characterizing success, failure, and error patterns. Experimental validation confirms >99.9% accuracy under the derived conditions, precisely delineating the performance boundaries of p-Conv.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

In this paper, we propose a new theoretical approach to Explainable AI. Following the Scientific Method, this approach consists in formulating on the basis of empirical evidence, a mathematical model to explain and predict the behaviors of Neural Networks. We apply the method to a case study created in a controlled environment, which we call Prime Convolutional Model (p-Conv for short). p-Conv operates on a dataset consisting of the first one million natural numbers and is trained to identify the congruence classes modulo a given integer $m$. Its architecture uses a convolutional-type neural network that contextually processes a sequence of $B$ consecutive numbers to each input. We take an empirical approach and exploit p-Conv to identify the congruence classes of numbers in a validation set using different values for $m$ and $B$. The results show that the different behaviors of p-Conv (i.e., whether it can perform the task or not) can be modeled mathematically in terms of $m$ and $B$. The inferred mathematical model reveals interesting patterns able to explain when and why p-Conv succeeds in performing task and, if not, which error pattern it follows.

Problem

• Develops a mathematical model for Explainable AI.
• Explains Neural Network behavior using empirical evidence.
• Identifies patterns in task success and error.

Innovation

• Prime Convolutional Model for Explainable AI
• Mathematical modeling of Neural Network behaviors
• Empirical validation with congruence classes