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.
The studies have established the possibility of reducing computational complexity, higher productivity of minimization of the Boolean functions in the class of expanded normal forms of the Sheffer algebra functions by the method of image transformations. Expansion of the method of image transformations to the minimization of functions of the Sheffer algebra makes it possible to identify new algebraic rules of logical transformations. Simplification of the Sheffer functions on binary structures of the 2-(n, b)-designs features exceptional situations. They are used both when deriving the result of simplification of functions from a binary matrix and introducing the Sheffer function to the matrix. It was shown that the expanded normal form of the n-digit Sheffer function can be represented by binary sets or a matrix. Logical operations with the matrix structure provide the result of simplification of the Sheffer functions. This makes it possible to concentrate the principle of minimization within the truth table of a given function and do without auxiliary objects, such as Karnaugh map, Weich diagrams, coverage tables, etc. Compared with the analogs of minimizing the Sheffer algebra functions, the method under the study makes the following to be possible:-reduce algorithmic complexity of minimizing expanded normal forms of the Sheffer functions (ENSF-1 and ENSF-2);-increase the productivity of minimizing the Sheffer algebra functions by 100-150 %;-demonstrate clarity of the process of mi nimizing the ENSF-1 or ENSF-2;-ensure self-sufficiency of the method of image transformations to minimize the Sheffer algebra functions by introducing the tag of mini mum function and minimization in the complete truth table of the ENSF-1 and ENSF-2. There are reasons to assert that application of the method of image transformations to the minimization of the Sheffer algebra functions brings the problem of minimization of the ENSF-1 and ENSF-2 to the level of a wellstudied problem in the class of disjunctiveconjunctive normal forms (DCNF) of Boolean functions
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