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

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 symmetrical Boolean functions in the main and polynomial bases. Prospective reserves in the analytical method were identified, such as simplification of polynomial function conjuncterms using the created equivalent transformations based on the method of inserting the same conjuncterms followed by the operation of super-gluing the variables. The method of figurative transformations was extended to the process of minimizing the symmetrical Boolean functions with the help of algebra in terms of rules for simplifying the functions of the main and polynomial bases and developed equivalent transformations of conjuncterms. It was established that the simplification of symmetric Boolean functions by the method of figurative transformations is based on a flowchart with repetition, which is the actual truth table of the assigned function. This is a sufficient resource to minimize symmetrical Boolean functions that makes it possible to do without auxiliary objects, such as Karnaugh maps, cubes, etc. The perfect normal form of symmetrical functions can be represented by binary matrices that would represent the terms of symmetrical Boolean functions and the OR or XOR operation for them. The experimental study has confirmed that the method of figurative transformations that employs the 2-(n, b)-design, and 2-(n, x/b)-design combinatorial systems improves the efficiency of minimizing symmetrical Boolean functions. Compared to analogs, this makes it possible to enhance the productivity of minimizing symmetrical Boolean functions by 100‒200 %. There are grounds to assert the possibility of improving the effectiveness of minimizing symmetrical Boolean functions in the main and polynomial bases by the method of figurative transformations. This is ensured, in particular, by using the developed equivalent transformations of polynomial function conjuncterms based on the method of inserting similar conjuncterms followed by the operation of super-gluing the variables.

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

“…The mathematical apparatus for minimizing Boolean functions by the method of figurative transformations was considered in works [10,12,13,16,17,23,24], etc.…”

“…The first matrix of expression (25) represents the truth table of the partially defined DNF of the function f(x 1 , x 2 , x 3 , x 4 ) (24).…”

“…which contains one less inversion compared to the simplification using VBA MS Excel [18]. We note that the undefined sets of variables 1, 2, 10 were not used during the minimization of function (24). This ultimately reduced the overall complexity of simplifying the given partially defined function (24).…”