Neural networks as fuzzy logic formulas

2026-05-04

AI Summary

This work addresses a critical gap in the theoretical understanding of standard neural networks equipped with ReLU activations and rational weights, for which no logical characterization previously existed, thereby limiting insights into their expressive power. The paper introduces Rational Pavelka Logic (RPL) and its extensions, integrating them with a fragment of LΠ½ fuzzy logic to rigorously formalize the computational behavior of such networks as fuzzy logical formulas. This framework accommodates activation values over arbitrary real numbers and generalizes to ReLU-based networks operating within generalized polynomial rings. By establishing an exact equivalence between ReLU neural networks and two classical systems of fuzzy logic, the study provides a mathematically rigorous foundation for analyzing both the logical interpretability and expressive capabilities of these widely used models.

Abstract

Neural networks are a fundamental aspect of modern artificial intelligence, playing a key role in various important machine learning architectures including transformers and graph neural networks. Recently, logical characterisations have been used to study the expressive power of many machine learning architectures, but logical characterisations of plain neural networks have received less attention. In this paper, we provide fuzzy logic characterisations of rational-weight ReLU-activated neural networks via two well-established fuzzy logics: Rational Pavelka Logic RPL (and extensions thereof) and (fragments of) $\mathit{L Π} \frac{1}{2}$. The activation values of the neural networks are allowed to be arbitrary real numbers. We also provide fuzzy logic characterisations of a generalised polynomial ring over $\mathbb{Q}$ in countably many variables where the use of the ReLU-function is permitted.

Problem

• neural networks
• fuzzy logic
• ReLU activation
• logical characterization
• expressive power

Innovation

• fuzzy logic
• ReLU neural networks
• Rational Pavelka Logic
• LΠ1/2 logic
• logical characterization