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 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. We also obtain translations between the rule-based logic for Boolean networks, the diamond-free fragment of modal substitution calculus and a class of recursive Boolean circuits where the number of input and output gates match.

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。我们考虑通用场景：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些神经网络。特别地，我们证明了神经网络规模与相应布尔规则公式之间存在多项式关系。事实上，从布尔规则到神经网络的转换中，规模增长仅为线性。此外，我们分析了转换过程中运行时间的延迟：从神经网络到布尔规则的转换中，时间延迟与神经网络规模呈多对数关系，与运行时间呈线性关系；而在反向转换中，时间延迟与两者均呈线性关系。我们还建立了布尔网络基于规则的逻辑、模态替换演算的无菱形片段以及一类输入输出门数量匹配的递归布尔电路之间的转换关系。