Abstract
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 minimizing 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 minimum 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 well-studied problem in the class of disjunctive-conjunctive normal forms (DCNF) of Boolean functions
Highlights
The entire history of digital circuits is based on the choice of logical basis and optimization of functions on this basis.Webb and Sheffer bases attract attention because they consist of one function, that is, they are monofunctional and implemented by transistor circuits
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
The study objective is to extend the method of image transformations to the minimization of the Boolean functions in the class of perfect normal forms of the Sheffer functions (ENSF-1 and Expanded normal Sheffer form 2 (ENSF-2))
Summary
The entire history of digital circuits is based on the choice of logical basis and optimization of functions on this basis. Almost all known methods of minimizing logic circuits from Karnaugh maps to Espresso algorithms, give results based on {AND, OR, NOT} as well [4, 5]. After such minimization, special algorithms replace elements of the {AND, OR, NOT} basis with elements of the {NOR, NAND} basis. Since the Sheffer basis belongs to the field of optimization of logical functions [10], the studies aimed at improvement of such factors as:.
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