Developing an algorithm to minimize boolean functions for the visual-matrix form of the analytical method

Abstract

This research has established the possibility of improving the effectiveness of the visual-matrix form of the analytical Boolean function minimization method by identifying reserves in a more complex algorithm for the operations of logical absorption and super-gluing the variables in terms of logical functions. An improvement in the efficiency of the Boolean function minimization procedure was also established, due to selecting, according to the predefined criteria, the optimal stack of logical operations for the first and second binary matrices of Boolean functions. When combining a sequence of logical operations using different techniques for gluing variables such as simple gluing and super-gluing, there are a small number of cases when function minimization is more effective if an operation of simply gluing the variables is first applied to the first matrix. Thus, a short analysis is required for the primary application of operations in the first binary matrix. That ensures the proper minimization efficiency regarding the earlier unaccounted-for variants for simplifying the Boolean functions by the visual-matrix form of the analytical method. For a series of cases, the choice of the optimal stack is also necessary for the second binary matrix. The experimental study has confirmed that the visual-matrix form of the analytical method, whose special feature is the use of 2-(n, b)-design and 2-(n, x/b)-design systems in the first matrix, improves the process efficiency, as well as the reliability of the result of Boolean function minimization. This simplifies the procedure of searching for a minimal function. Compared to analogs, that makes it possible to improve the productivity of the Boolean function minimization process by 100‒200 %. There is reason to assert the possibility of improving the efficiency of the Boolean function minimization process by the visual-matrix form of the analytical method, through the use of more complex logical operations of absorbing and super-gluing the variables. Also, by optimally combining the sequence of logical operations of super-gluing the variables and simply gluing the variables, based on the selection, according to the established criteria, of the stack of logical operations in the first binary matrix of the assigned function

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

“…It was noted that the reported method for synthesizing symmetrical functions is the first, which is aimed at reducing the depth of logical schemes generated for symmetrical Boolean functions. The experimental results demonstrated that the approach in question reduces the depth of the final implementation of the function to 25.93 % compared to other methods of synthesis of symmetrical functions.…”

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

“…This determines the need to investigate equivalent figurative transformations in order to minimize logic functions on the Reed-Muller basis. In particular, the peculiarities of relatively complex algorithms of simplification of functions with the procedure of inserting the same conjuncterms of polynomial functions followed by the operation of super-gluing the variables, a stack of logical operations for the first binary matrix of a polynomial function [9], ways to simplify arbitrary functions in the Reed-Muller basis.…”

“…This is a sufficient resource to minimize functions and makes it possible to do without auxiliary objects, such as Karnaugh maps, Weich diagrams, acyclic graph, non-directed graph, coverage tables, cubes, etc. The clarity of 2-dimensional binary matrices allows one to manually simplify Boolean functions (using a mathematical editor, such as MathType 7.4.0) within up to 64 input variables [9] for PESOP (DCNF) representation of the function.…”

“…The selection of the stack of logic operations [9] for ESOP functions is demonstrated by the following example.…”

“…The evolution of the visual-matrix form of the analytical method of simplification of logic functions is the result of continuous optimization, in particular [9]. In this regard, theoretical studies on minimizing ESOP of Boolean functions remain relevant, in particular, to improve the following factors:…”