Boolean functions minimization by the method of figurative transformations

Abstract

The object of research is the method of figurative transformations for Boolean functions minimization. One of the most problematic places to minimize Boolean functions is the complexity of the minimization algorithm and the guarantee of obtaining a minimal function. During the study, the method of equivalent figurative transformations was used, which is based on the laws and axioms of the algebra of logic; minimization protocols for Boolean functions that are used when the truth table of a given function has a complete binary combinatorial system with repetition or an incomplete binary combinatorial system with repetition. A reduction in the complexity of the minimization process for Boolean functions is obtained, new criteria for finding minimal functions are established. This is due to the fact that the proposed method of Boolean functions minimization has a number of peculiarities of solving the problem of finding minimal logical functions, in particular: – mathematical apparatus of the block diagram with repetition makes it possible to obtain more information about the orthogonality, adjacency, uniqueness of the truth table blocks; – equivalent figurative transformations due to the greater information capacity are capable of replacing verbal procedures of algebraic transformations; – result of minimization is estimated based on the sign of the minimum function; – minimum DNF or CNF functions are obtained regardless of the given normal form of the logical function, which means that it is necessary to minimize the given function for two normal forms — DNF and CNF using the full truth table; This ensures that it is possible to obtain an optimal reduction in the number of variables of a given function without losing its functionality. The effectiveness of the use of equivalent figurative transformations for Boolean functions minimization is demonstrated by examples of minimization of functions borrowed from other methods for the purpose of comparison. Compared with similar well-known methods of Boolean functions minimization, this provides: – less complexity of the minimization procedure for Boolean functions; – guaranteed Boolean functions minimization; – self-sufficiency of the specified method of Boolean functions minimization due to the introduction of features of the minimal function and minimization of two normal forms – DNF and CNF on the complete truth table of a given Boolean function