Descriptive complexity for neural networks via Boolean networks

Abstract We investigate the expressive power of neural networks from the point of view of descriptive complexity. We study neural networks that use floating-point numbers and piecewise polynomial activation functions from two perspectives: (i) the general scenario where neural networks run for an unlimited number of computation steps and have unrestricted topologies, and (ii) classical feedforward neural networks that have the topology of layered acyclic graphs and run for only a constant number of computation steps. We characterize these neural networks via Boolean networks formalized via a recursive rule-based logic. In particular, we show that the sizes of the neural networks and the corresponding Boolean rule formulae are polynomially related. In fact, in the translation from Boolean rules to neural networks, the blow-up is only linear. Our translations result in a time delay, defined as the number of computation steps that the output-object of the translation (e.g. a neural network or Boolean rule formula) uses to simulate a single computation step of the input-object. In the translation from neural networks to Boolean rules, the time delay of the resulting formula is polylogarithmic in the size of the neural network. In the converse translation, the time delay of the neural network is linear in the formula size. Ultimately, we obtain translations between neural networks, Boolean networks, the diamond-free fragment of modal substitution calculus and a class of recursive Boolean circuits. Our translations offer a method, for almost any activation function F, of translating any neural network in our setting into an equivalent neural network that uses F at each node. This even includes linear activation functions, which is possible due to using floats rather than actual reals!