Solving the parity n problem and other nonlinearly separable problems using a single universal binary neuron

Abstract

A universal binary neuron (UBN) operates with the complex-valued weights and the complex-valued activation function, which is the function of the argument of the weighted sum. This makes possible the implementation of the nonlinearly separable (nonthreshold) Boolean functions on the single neuron. Hence the functionality of the UBN is incompatibly higher than the functionality of the traditional perceptron, because this neuron can implement the nonthreshold Boolean functions. The UBN is closely connected with the discrete-valued multi-valued neuron (MVN). This is also a neuron with the complex-valued weights and the complexvalued activation function, which is the function of the argument of the weighted sum. A close relation of the MVN and UBN and of the multiple-valued threshold functions and P-realizable Boolean functions is considered in this paper. A modified learning algorithm for the UBN is presented. It is shown that such classical nonlinearly separable problems as the XOR and Parity n can be easily solved using a single UBN, without any network.