Topological equivalence classification and enumeration of n-input linearly separable Boolean functions

Abstract

A perceptron, as an “artificial neuron”, plays a crucial role in the research of neural networks. As a mathematical model of linearly separable Boolean functions, perceptron neurons can realize any nonlinearly separable Boolean function. Linearly separable Boolean functions can be divided into a smaller number of the same functional classes. These functional classes are equivalent to the core basic units (AND, NOT, OR, and other components) that make up the circuit. Therefore, the research on the classification problem of linearly separable Boolean functions is very important. In this study, a novel algorithm, called a topological equivalence classification algorithm, is proposed for classifying balanced linearly separable Boolean functions. By the proposed algorithm, the total number of the topological equivalence classes and the number of balanced linearly separable Boolean functions in each of the topological equivalence classes are obtained. According to the one-to-one correspondence between n-input balanced linearly separable Boolean functions and n-1-input linearly separable Boolean functions, we get the topological classification of linearly separable Boolean functions. As for n=9, the accurate results of classification and counting the number of linearly separable Boolean functions are obtained: There are 53,063,448 topological equivalence classes, and the total number of linearly separable Boolean functions is 144,380,202,286,068,288.