Research of 5-bit boolean functions minimization protocols by combinatorial method

Abstract

The object of research is a combinatorial method of 5-bit Boolean functions minimization. One of the most problematic places for Boolean functions minimization is the complexity of the minimization algorithm and the guarantee of obtaining a minimal function.Minimization protocols of the 5-bit Boolean functions are used in the course of the research, which are used when the structure of the truth table of a given function has a complete binary combinatorial system with repetition or an incomplete binary combinatorial system with repetition. The operational properties of the protocols for 5-bit Boolean functions minimization are based on the laws and axioms of the algebra of logic.A reduction in the complexity of the process of 5-bit Boolean functions minimization by combinatorial method is obtained, increasing the probability of guaranteed 5-bit Boolean functions minimization. This is due to the fact that the proposed method of 5-bit Boolean functions minimization has a number of features to solve the problem of minimizing the logical function, in particular:the mathematical apparatus of the block diagram with repetition makes it possible to obtain more information on the orthogonality, contiguity, uniqueness of truth table blocks;equivalent transformations by graphic images in the form of two-dimensional matrices due to the greater information capacity can with effect replace the verbal procedures of algebraic transformations;minimization protocols for 5-bit Boolean functions constitute a protocol library for the process of 5-bit Boolean functions minimization as standard procedures, so the use of a separate protocol for variables of 5-bit Boolean functions is reduced to carrying out one algebraic transformation.Thanks to this, it is possible to obtain an optimal reduction in the number of variable functions without losing its functionality. The effectiveness of the application of minimization protocols for the 5-bit Boolean functions of the combinatorial method is demonstrated by examples of minimization of functions taken from the work of other authors for the purpose of comparison.In comparison with similar known methods of Boolean functions minimization, this ensures:less complexity of the process of 5-bit Boolean functions minimization;an increase in the probability of guaranteed 5-bit Boolean functions minimization;improvement of the algebraic method of Boolean function minimization due to the tabular organization of the combinatorial method, the introduction of the image-transformation apparatus and the minimization protocols.