We provide a refined characterization of the super-Turing computational power of analog, evolving, and stochastic neural networks based on the Kolmogorov complexity of their real weights, evolving weights, and real probabilities, respectively. First, we retrieve an infinite hierarchy of classes of analog networks defined in terms of the Kolmogorov complexity of their underlying real weights. This hierarchy is located between the complexity classes $\mathbf{P}$ and $\mathbf{P/poly}$. Then, we generalize this result to the case of evolving networks. A similar hierarchy of Kolomogorov-based complexity classes of evolving networks is obtained. This hierarchy also lies between $\mathbf{P}$ and $\mathbf{P/poly}$. Finally, we extend these results to the case of stochastic networks employing real probabilities as source of randomness. An infinite hierarchy of stochastic networks based on the Kolmogorov complexity of their probabilities is therefore achieved. In this case, the hierarchy bridges the gap between $\mathbf{BPP}$ and $\mathbf{BPP/log^*}$. Beyond proving the existence and providing examples of such hierarchies, we describe a generic way of constructing them based on classes of functions of increasing complexity. For the sake of clarity, this study is formulated within the framework of echo state networks. Overall, this paper intends to fill the missing results and provide a unified view about the refined capabilities of analog, evolving and stochastic neural networks.

翻译：我们基于模拟神经网络中实值权重、进化神经网络中进化权重以及随机神经网络中真实概率的柯尔莫哥洛夫复杂性，对其超图灵计算能力进行了精炼刻画。首先，我们根据底层实值权重的柯尔莫哥洛夫复杂性，重新构建了一个无穷层次的模拟网络类别体系。该层次结构位于复杂性类$\mathbf{P}$与$\mathbf{P/poly}$之间。随后，我们将此结果推广至进化网络场景，获得了基于柯尔莫哥洛夫复杂性的进化网络复杂性类同构层次结构，该层级同样介于$\mathbf{P}$与$\mathbf{P/poly}$之间。最后，我们将这些结果扩展至采用真实概率作为随机性来源的随机网络情形，从而建立了基于概率柯尔莫哥洛夫复杂性的无穷层次随机网络体系。在此情形下，该层次结构桥接了$\mathbf{BPP}$与$\mathbf{BPP/log^*}$之间的空白区间。除证明此类层次结构的存在性并提供实例外，我们描述了基于递增复杂性函数类构建这些层次结构的通用方法。为清晰起见，本研究以回声状态网络框架进行阐述。总体而言，本文旨在填补相关研究空白，并为模拟、进化与随机神经网络的精炼能力提供统一视角。