The algorithm for minimizing Boolean functions using a method of the optimal combination of the sequence of figurative transformations

Riznyk, Volodymyr; Solomko, Mykhailo; Tadeyev, Petro; Nazaruk, Vitalii; Zubyk, Liudmyla; Волошин, Володимир

doi:10.15587/1729-4061.2020.206308

Cited by 7 publications

(23 citation statements)

References 9 publications

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“…Paper [12] considers the consistent alternation of logical operations of super-gluing the variables (if such an operation is possible) and simple gluing of variables in the first matrix of the Boolean function, which ensures the efficiency of minimization and is the basis for the algorithmization of the analytical method. The authors demonstrated an example of minimizing the 4-bit Boolean function when the consistent use of these logical operations is not always optimal in terms of the procedure effectiveness.…”

Section: X X X X X X X X X X X Xmentioning

confidence: 99%

“…The evolution of the analytical method of simplification of logical functions and its algorithmization is the result of continuous optimization, so it has remained relevant to undertake research aimed, in particular, at making an update to the minimization algorithm by the analytical method, consi dered in work [12]. That would render proper efficiency to the previously unaccounted-for variants for simplifying Boolean functions by the analytical method, in particular in the class of PDNF and PKNF representations, as well as make it possible to optimize the cost of technology of Boolean function minimization by the analytical method.…”

Section: X X X X X X X X X X X Xmentioning

confidence: 99%

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Developing an algorithm to minimize boolean functions for the visual-matrix form of the analytical method

Solomko¹

2021

EEJET

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This research has established the possibility of improving the effectiveness of the visual-matrix form of the analytical Boolean function minimization method by identifying reserves in a more complex algorithm for the operations of logical absorption and super-gluing the variables in terms of logical functions. An improvement in the efficiency of the Boolean function minimization procedure was also established, due to selecting, according to the predefined criteria, the optimal stack of logical operations for the first and second binary matrices of Boolean functions. When combining a sequence of logical operations using different techniques for gluing variables such as simple gluing and super-gluing, there are a small number of cases when function minimization is more effective if an operation of simply gluing the variables is first applied to the first matrix. Thus, a short analysis is required for the primary application of operations in the first binary matrix. That ensures the proper minimization efficiency regarding the earlier unaccounted-for variants for simplifying the Boolean functions by the visual-matrix form of the analytical method. For a series of cases, the choice of the optimal stack is also necessary for the second binary matrix. The experimental study has confirmed that the visual-matrix form of the analytical method, whose special feature is the use of 2-(n, b)-design and 2-(n, x/b)-design systems in the first matrix, improves the process efficiency, as well as the reliability of the result of Boolean function minimization. This simplifies the procedure of searching for a minimal function. Compared to analogs, that makes it possible to improve the productivity of the Boolean function minimization process by 100‒200 %. There is reason to assert the possibility of improving the efficiency of the Boolean function minimization process by the visual-matrix form of the analytical method, through the use of more complex logical operations of absorbing and super-gluing the variables. Also, by optimally combining the sequence of logical operations of super-gluing the variables and simply gluing the variables, based on the selection, according to the established criteria, of the stack of logical operations in the first binary matrix of the assigned function

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Section: X X X X X X X X X X X Xmentioning

confidence: 99%

Section: X X X X X X X X X X X Xmentioning

confidence: 99%

Developing an algorithm to minimize boolean functions for the visual-matrix form of the analytical method

Solomko¹

2021

EEJET

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“…The mathematical apparatus of image transformations was considered in [10,19,20,22,24]. Hermeneutics of logic operations with binary structures provides sufficient didactics of simplification of the Boolean functions including the class of expanded normal forms of functions of the Sheffer algebra.…”

Section: Discussion Of the Results Obtained In Minimization Of The Shmentioning

confidence: 99%

“…/ , so the «false» value of the «Sheffer stroke» operation is only obtained when both arguments x 1 and x 2 take the «true» value (Table 1). Table 1 Truth table of the «Sheffer stroke» operation Since the Sheffer basis belongs to the field of optimization of logical functions [10], the studies aimed at improvement of such factors as:…”

Section: Fig 1 Implementation Of the Elements Of The Basic Basis {Nmentioning

confidence: 99%

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Implementation of the method of image transformations for minimizing the Sheffer functions

Solomko¹,

Хомюк²,

Ivashchuk³

et al. 2020

EEJET

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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 mi nimizing 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 mini mum 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 wellstudied problem in the class of disjunctiveconjunctive normal forms (DCNF) of Boolean functions

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Information and logical transformations in Schaeffer and Pierce Bases in Maple

Olenev,

Potekhina,

Khabarov

et al. 2024

ITM Web Conf.

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One of the tasks of synthesis and analysis of the functioning of combinational devices is the representation of a Boolean function in various bases. The absolute completeness of the use and implementation of various transformations of Boolean functions is impossible without the implementation of transformations into universal bases of Schaeffer (AND-NOT) or Webb (OR-NOT). The Logic library of the Maple computer algebra system allows you to use only the operations &nand (Schaeffer basis) and &nor (Webb basis) to build a model of a digital device. Therefore, there is a need to expand the functionality of the Logic library of the Maple computer algebra system by representing Boolean functions in Schaeffer and/or Pierce (Webb) bases. In this regard, the authors of the article have developed procedures that allow the representation of disjunctive or conjunctive normal forms in the Schaeffer basis or in the Webb basis. The development was carried out taking into account the existing data processing functions and procedures. The article describes the technology of extending the Logic library of the Maple computer algebra system by developed procedures that implement the transformation of a given or received Boolean function in Schaeffer or Webb bases.

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The algorithm for minimizing Boolean functions using a method of the optimal combination of the sequence of figurative transformations

Cited by 7 publications

References 9 publications

Developing an algorithm to minimize boolean functions for the visual-matrix form of the analytical method

Developing an algorithm to minimize boolean functions for the visual-matrix form of the analytical method

Implementation of the method of image transformations for minimizing the Sheffer functions

Information and logical transformations in Schaeffer and Pierce Bases in Maple

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