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.

翻译：神经网络是现代人工智能的基础组成部分，在包括Transformer和图神经网络在内的多种重要机器学习架构中发挥关键作用。近年来，逻辑刻画已被用于研究许多机器学习架构的表达能力，但普通神经网络的逻辑刻画却较少受到关注。本文通过两种成熟的模糊逻辑——有理帕维尔卡逻辑RPL（及其扩展）与（片段化的）$\mathit{L Π} \frac{1}{2}$，对有理权重ReLU激活神经网络提供了模糊逻辑刻画。神经网络中的激活值允许为任意实数。此外，我们还对可数变量上基于$\mathbb{Q}$的广义多项式环（允许使用ReLU函数）提供了模糊逻辑刻画。