Zhegalkin Polynomial of a Multiary Sole Sufficient Operator

Abstract

Sole sufficient operators are of particular interest among functionally complete sets of Boolean functions. They have a wide range of applicability and are not limited to the binarity case. In this paper, we formulate conditions imposed on the Zhegalkin polynomial coefficients that are necessary and sufficient for the polynomial to correspond to a sole sufficient operator. The polynomial representation of constant-preserving Boolean functions is considered. It is shown that the properties of monotonicity and linearity do not need to be specifically considered when describing a sole sufficient operator. The concept of a dual remainder polynomial is introduced; the value of it allows one to determine the self-duality of a Boolean function. It is proven that a Boolean function preserving 0 and 1 or preserving neither 0 nor 1 is self-dual if and only if the dual remainder of the corresponding Zhegalkin polynomial is 0 for any sets of values of the function variables. The system of leading coefficients is obtained based on this fact. The solution of the system makes it possible to formulate a criterion for the self-duality of a Boolean function represented by a Zhegalkin polynomial, which imposes necessary and sufficient conditions on the polynomial coefficients. Thus, it is shown that Zhegalkin polynomials are a rather convenient tool for studying precomplete classes of Boolean functions.