A New Method of the Logical Functions Minimization in the Polynomial Set-Theoretical Format. «Handshaking» Procedure

Abstract

A new minimization method of logic functions of n variables in polynomial set-theoretical format has been considered. The method based on the so-called “handshaking” procedure. This procedure reflects the iterative polynomial extension of two conjuncterms of different ranks, the Hamming distance between which can be arbitrary. The advantages of the suggested method are illustrated by the examples.

Implementing the method of figurative transformations to minimize partially defined Boolean functions

“…Generalized rules for simplifying conjuncterms in a polynomial theoretical-set format are discussed in [7]. These rules are based on the proposed theorems for different initial conditions for the transformation of paired conjuncterms, the Hemming distance between which can be arbitrary.…”

“…The vast majority of functions and their systems minimized by the proposed method in [7] showed the best results. This is due to the fact that conjuncterms with a Hemming distance d ≥ 3 are involved in the transformation process.…”

“…Note that the theory is focused on conclusions about the inexpediency of further minimization in the polynomial format of Boolean functions if the distance between an arbitrary pair of its conjuncterms was d ≥ 3. When applying the theorems [7] to the function f, which is given not by two, but by a larger number of conjuncterms of different ranks, between arbitrary pairs for which there is a different distance d, it is quite possible to further simplify it. This is explained by the fact that the set of transformed PTMFs Y ⊕ , with which the selected pair with a distance of d ≥ 3 is replaced, may contain elements that, with other elements of the given function f, will form pairs with a distance of d < 3.…”

“…This is explained by the fact that the set of transformed PTMFs Y ⊕ , with which the selected pair with a distance of d ≥ 3 is replaced, may contain elements that, with other elements of the given function f, will form pairs with a distance of d < 3. Moreover, despite the increase in the power of the newly formed set, after applying the corresponding rules of theorems [7] to pairs with small d, it is possible to obtain a minimal PTMF Y ⊕ . As a result, the chance of effectively simplifying the set of conjuncterms increases.…”

“…With an increase in the number of indefinite sets, the set of fully defined Boolean functions increases significantly. For seven, for example, indefinite sets of variables, there are 2 7 = 128 different ways of binary redefinition of partially defined Boolean functions, and, therefore, the complexity of the function minimization procedure will increase.…”