Neural networks as fuzzy logic formulas
For researchers in logic and AI, this work bridges neural networks and fuzzy logic, offering a new theoretical perspective on the expressive power of neural networks.
The paper provides fuzzy logic characterizations of rational-weight ReLU-activated neural networks using Rational Pavelka Logic and fragments of LΠ½, enabling the representation of neural network computations as fuzzy logic formulas.
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.