Periodic Symmetric Functions with Feed-Forward Neural Networks

This technical report presents a new theoretical approach to the problem of switching networks synthesis with McCulloch-Pitts feed-forward neural networks. It is shown that any n-inputs periodical symmetric Boolean function Fp with the period T and the first positive transition at x = a can be implemented with a 1 + dlog n a T e depth and size network both measured in term of neurons, when a period contains two transitions. It can be implemented with a t+dlog n a T e depth and size network when a period contains more than two transitions, where t is the number of neural elements necessary to implement the restriction of Fp to the first period, i.e. the input interval [0; T ]. An asymptotic bound of O(logn) for the network (for both size and depth) is also derived for symmetric Boolean functions that can be decomposed in l periodic symmetric Boolean sub-functions.