Neural networks as fuzzy logic formulas
神经网络作为模糊逻辑公式
Damian Heiman, Antti Kuusisto, Esko Turunen
AI总结 本文通过Rational Pavelka逻辑及其扩展,为有理权重ReLU激活的神经网络提供了模糊逻辑刻画,并推广到允许任意实数值激活的广义多项式环。
详情
神经网络是现代人工智能的一个基本方面,在包括Transformer和图神经网络在内的各种重要机器学习架构中扮演着关键角色。最近,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受到的关注较少。在本文中,我们通过Rational Pavelka逻辑($\mathrm{RPL}$)及其扩展$\mathrm{RPL}(\odot)_{\leq 1}$,以及$\mathit{L \Pi} \frac{1}{2}$的两个片段$\mathit{L \Pi} \frac{1}{2}(\rightarrow_{P}^-)_{\leq 1}$和$\mathit{L \Pi} \frac{1}{2}(\odot^-, \rightarrow_{P}^-)$,为有理权重ReLU激活的神经网络提供了模糊逻辑刻画。神经网络的激活值允许为任意实数。我们还通过模糊逻辑$\mathrm{RPL}(\odot)$和$\mathit{L \Pi} \frac{1}{2}$的一个片段$\mathit{L \Pi} \frac{1}{2}(\rightarrow_{P}^-)$,为可数多个变量上允许使用ReLU函数的广义多项式环$\mathbb{Q}$提供了模糊逻辑刻画。
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 Rational Pavelka logic ($\mathrm{RPL}$) and an extension of $\mathrm{RPL}$ called $\mathrm{RPL}(\odot)_{\leq 1}$, as well as two fragments of $\mathit{L Π} \frac{1}{2}$ called $\mathit{L Π} \frac{1}{2}(\rightarrow_{P}^-)_{\leq 1}$ and $\mathit{L Π} \frac{1}{2}(\odot^-, \rightarrow_{P}^-)$. 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 via the fuzzy logic $\mathrm{RPL}(\odot)$ and a fragment of $\mathit{L Π} \frac{1}{2}$ called $\mathit{L Π} \frac{1}{2}(\rightarrow_{P}^-)$.