Universal Perceptron Theory on Decomposition Structures of n-Bit Boolean Functions

Abstract

By using the universal perceptron theory, a decomposition algorithm called improved distance and DNA-like decomposition algorithm (ID-DNA-like DA) is proposed to investigate the linear decomposition structures of [Formula: see text]-bit Boolean functions. The specific linear decomposition structure of an [Formula: see text]-bit Boolean function is generated by the proposed decomposition algorithm ([Formula: see text]). Especially, the linear decomposition structure of 4-bit Boolean function is shown to have the following properties: the total number of 4-bit Boolean functions is 65[Formula: see text]536 and the number of linearly separable Boolean functions is 1882; the number of 2-length decomposition Boolean functions is not less than 41[Formula: see text]366; the number of 3-length decomposition Boolean functions does not exceed 22[Formula: see text]286; and the number of 4-length decomposition Boolean functions is only 2. At the same time, it is shown that the decomposition length of [Formula: see text]-bit Boolean function does not exceed [Formula: see text] [Formula: see text]. In addition, a concept of the [Formula: see text]-bit Boolean function decomposition structure figure is proposed. By decomposition calculation, the related results of decomposition structure figures show that there is some symmetry with a strong self-similar fractal feature.