Boolean Soft Computing by Non-linear Neural Networks With Hyperincursive Stack Memory

Abstract

This paper is a review of a new theoretical basis for modelling neural Boolean networks by non-linear digital equations. With real numbers, soft Boolean tables can be generated. With integer numbers, these digital equations are Heaviside fixed functions in the framework of the threshold logic. These can represent non-linear neurons which can be split very easily into a set of McCulloch and Pitts formal neurons with hidden neurons. It is demonstrated that any Boolean tables can be very easily represented by such neural networks where the weights are always either an activation weight +1 or an inhibition weight -1, with integer threshold. The parity problem is fully solved by a fractal neural network based on XOR. From a feedback of the hidden neurons to the inputs in a XOR non-linear equations, it is showed that the neurons compete with each other. Moreover, the feedback of the output to the inputs for a XOR non-linear neuron gives rise to fractal chaos. A model of a stack memory can be designed from such a chaos map Binary digits are memorised by folding to a real variable by an anti-chaotic hyperincursive process. The retrieval of these data is computed by an incursive chaotic map from the last value of the variable. Incursion is an extension of recursion for which each iterate is computed in function of variables not only defined in the past and the present time by also in the future. Hyperincursion is an incursion generating multiple iterates at each step. The basic map is the PearlVerhulst one in the zone of fractal chaos. The hyperincursive memory realises a coding of the input binary message under the form similar to the Gray code. This is based on a soft exclusive OR equation mixing binary digits with real numbers.