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 (non-threshold) 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 non-threshold 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 complex-valued 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.