We investigate the descriptive complexity of a class of neural networks with unrestricted topologies and piecewise polynomial activation functions. We consider the general scenario where the running time is unlimited and floating-point numbers are used for simulating reals. We characterize a class of these neural networks with a rule-based logic for Boolean networks. In particular, we show that the sizes of the neural networks and the corresponding Boolean rule formulae are polynomially related. In fact, in the direction from Boolean rules to neural networks, the blow-up is only linear. We also analyze the delays in running times due to the translations. In the translation from neural networks to Boolean rules, the time delay is polylogarithmic in the neural network size and linear in time. In the converse translation, the time delay is linear in both factors.

翻译：我们研究了一类具有无限制拓扑结构和分段多项式激活函数的神经网络的描述复杂度。我们考虑了通用场景，其中运行时间不受限制，且使用浮点数模拟实数。我们通过一种基于规则的布尔网络逻辑来刻画这类神经网络的特性。具体而言，我们证明了神经网络的大小与对应的布尔规则公式的大小之间存在多项式相关关系。实际上，从布尔规则到神经网络的转换过程中，规模增长仅为线性。我们还分析了由于转换导致的运行时间延迟。在从神经网络到布尔规则的转换中，时间延迟在神经网络规模上呈多对数关系，在时间上呈线性关系。而在反向转换中，时间延迟在两个因素上均呈线性关系。