О связи функциональных систем полиномов и арифметических полиномов, представляющих системы булевых функций

Abstract

The theory of functional systems is one of the basic fields of mathematical cybernetics. A functional system is a pair (P, O), in which P is the system carrier and O is the totality of operations specified in P. That is, a functional system is an algebra the elements of which are functions, and operations in this algebra correspond to the rules according to which the functions are constructed from each other. Functional systems involving a superposition operation (renaming and identifying the variables, and substituting functions for the variables of another function) are considered to be the conventional model objects of the theory. A functional system of polynomials with integer coefficients is investigated. The connection between a functional system of polynomials with integer coefficients and the arithmetic polynomials introduced by V.D. Malyugin to perform parallel logic computations is considered. Linear polynomials with integer coefficients are analyzed. The set of all such functions is denoted as L(Z) and is considered as a subset in the broader set P(Z) of functions represented by arbitrary-power polynomials with integer coefficients. Superposition (composition) operations are defined in L(Z) and P(Z). Arithmetic polynomials representing some systems of Boolean functions are given. It has been found that the set of arithmetic polynomials representing certain systems of Boolean functions do not coincide with P(Z). However, the closure with respect to the superposition operation of the set of arithmetic polynomials representing certain systems of Boolean functions coincides with P(Z). It is shown that the set of linear arithmetic polynomials representing some systems of Boolean functions does not coincide with L(Z). However, the closure with respect to the superposition operation of the set of linear arithmetic polynomials representing certain systems of Boolean functions coincides with L(Z).