Developing the minimization of a polynomial normal form of boolean functions by the method of figurative transformations

Abstract

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize Boolean functions on the Reed-Muller basis. Such potential prospects in the analytical method have been identified as a sequence in the procedure of inserting the same conjuncterms of polynomial functions followed by the operation of super-gluing the variables. The extension of the method of figurative transformations to the process of simplifying the functions of the polynomial basis involved the developed algebra in terms of the rules for simplifying functions in the Reed-Muller basis. It was established that the simplification of Boolean functions of the polynomial basis by a figurative transformation method is based on a flowchart with repetition, which is actually the truth table of the predefined function. This is a sufficient resource to minimize functions that makes it possible not to refer to such auxiliary objects as Karnaugh maps, Weich charts, cubes, etc. A perfect normal form of the polynomial basis functions can be represented by binary sets or a matrix that would represent the terms of the functions and the addition operation by module two for them. The experimental study has confirmed that the method of figurative transformations that employs the systems of 2-(n, b)-design, and 2-(n, x/b)-design in the first matrix improves the efficiency of minimizing Boolean functions. That also simplifies the procedure for finding a minimum function on the Reed-Muller basis. Compared to analogs, this makes it possible to enhance the performance of minimizing Boolean functions by 100‒200 %. There is reason to assert the possibility of improving the efficiency of minimizing Boolean functions in the Reed-Muller basis by a method of figurative transformations. This is ensured by using more complex algorithms to simplify logical expressions involving a procedure of inserting the same function terms in the Reed-Muller basis, followed by the operation of super-gluing the variables.

Implementation of the method of figurative transformations to minimizing symmetric Boolean functions

“…, ,1 with the transition to a mixed basis. Since function (1) is singular [26], let us dig into the Reed-Muller algebra. We obtain:…”

“…To simplify function (2), the procedure of inserting similar conjuncterms was applied to the polynomial normal form (PNF) with the following operation of super-gluing the variables [26].…”

“…The procedure for inserting similar conjuncterms with the following operation of super-gluing the variables [26] makes it possible to derive equivalent transformations of the normal polynomial form of Boolean functions. Some equivalent transformations of a normal polynomial form are derived as follows:…”

“…Algebraic expressions in the left column in Table 2 are singular [26], so the latter can be represented in the main basis ∨ ∧ ¬ { }…”

“…The prospect of the method of figurative transformations, as a descendant of the analytical method, regarding the proper minimization of symmetric logical functions in the Reed-Muller basis is the creation of the necessary algebra in terms of the rules for the equivalent transformation of polynomial functions [26]. As well as new developed equivalent transformations of conjuncterms in the polynomial normal form (PNF) based on the method of inserting similar conjuncterms with the following operation of super-gluing the variables (chapter 5.…”