A peculiarity of solving problems of minimization of Boolean functions

Abstract

Minimization of Boolean functions is mandatory for the construction of logic circuits of digital automata. The result of minimization, in general, can be not one, but several equivalent images of the Boolean function with the smallest number of variables and logical operations with them. However, a possible set of images of the minimal form of a Boolean function is not always are taken i nto account when solving minimization problems. Quite often, the result of minimization results in only one image, while considering that the problem is finally solved. Of course, such a solution is far from complete, it does not provide an opportunity to choose the optimal logic scheme of the digital automaton to be created. The purpose of the work is to justify the need to find all possible representations of the minimal form of the Boolean function. The task was solved by analyzing the minimization of an arbitrary Boolean function. The minimization was carried out analytically according to the Quine algorithm and coordinate using the Veitch diagram. In both cases, matching sets of images of the minimal form of the Boolean function are obtained, regardless of the chosen method of minimization. This testifies to the correctness of the solution to the minimization problem, the purpose of which is to find a set of images of the Boolean function to ensure the possibility of choosing the optimal solution when constructing a logic circuit of a digital automaton. It has been confirmed that the correct solution to the minimization problem is a mandatory image of not one possible function, but a set of images of all possible minimal forms of the Boolean function.