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

Refined Kolmogorov Complexity of Analog, Evolving and Stochastic Recurrent Neural Networks