Development of a non-standard system for simplifying boolean functions - Solomko, Mykhailo (Eastern-European Journal of Enterprise Technologies, 2024)
The object of this study is models of low-power digital logic circuits. The problem being solved is the effectiveness of the technique for simplifying Boolean functions to obtain optimal structures of logic circuits. A new theorem of a non-standard system of simplification of Boolean functions has been formulated, according to which in order to obtain a minimal function it will suffice to perform all non-redundant operations of simple and/or super-gluing of variables, which ultimately provides a minimal function in the main basis without using an implicant table. Thus, the problem of simplifying Boolean functions to the simplest normal equivalent is solved in one step. The interpretation of the result is that the properties of 2-(n, b)-design combinatorial systems make it possible to reproduce the definition of logical operations of super-gluing variables, to represent logical operations in a different way, and vice versa. This, in turn, ensures the establishment of the locations of equivalent transformations on the binary structure of the truth table and the implication of a systematic procedure for simplifying Boolean functions by an analytical method. Special feature of the results is that unambiguous identification of the locations of equivalent transformations is possible even when different intervals of the Boolean space containing the 2-(n, b)-design systems have common modules. It has been experimentally confirmed that the non-standard system improves the efficiency of simplifying Boolean functions, including partially defined ones, by 200–300 % compared to analogs. In terms of application, a non-standard system for simplifying Boolean functions will ensure the transfer of innovations to material production: from conducting fundamental research, expanding the capabilities of digital component design technology to organizing serial or mass production of novelties