DEVISING A METHOD OF FIGURATIVE TRANSFORMATIONS FOR MINIMIZING BOOLEAN FUNCTIONS IN THE IMPLICATIVE BASIS

Solomko M. та Batyshkina Iu. та Voitovych I. та Zubyk L. та Babych S. та Muzychuk K. (2020) DEVISING A METHOD OF FIGURATIVE TRANSFORMATIONS FOR MINIMIZING BOOLEAN FUNCTIONS IN THE IMPLICATIVE BASIS. Eastern-European Journal of Enterprise Technologies, 2020 (№6/4): 108. с. 32-47. ISSN 1729-3774

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Анотація

This paper reports a study that has established the possibility of reducing computational complexity while improving the productivity of simplification of Boolean functions in the class of perfect implied normal forms (PINF-1 and PINF-2) using a method of figurative transformations.
The method of figurative transformations has been expanded to cover the process of simplifying the functions of the implicative basis by using the developed algebra of the implicative basis in the form of rules that simplify the PINF-1 and PINF-2 functions of the implicative basis. A special feature in simplifying the functions of the implicative basis on the binary structures of 2-(n, b)-designs) is the use of analogs of perfect disjunctive normal forms (PDNF) and perfect conjunctive normal forms (PCNF) of Boolean functions. The specified forms of the functions define transformation rules for the functions of the implicative basis on binary structures.
It is shown that the perfect implicative normal form of n-place function of the implicative basis can be represented by the binary sets or a matrix.
Logical operations over the structure of the matrix ensure the result from simplifying the functions of the implicative basis. This makes it possible to focus the minimization principle within the truth table of the assigned function and avoid auxiliary objects such as Carnot map, Weich charts, etc.
The method under consideration makes it possible:
– to reduce the algorithmic complexity of PINF-1 and PINF-2 simplification; – to improve the performance of simplifying the functions of the implied basis by 100–200 %;
– to visualize the process of PINF-1 or PINF-2 minimization.
There is reason to argue that minimizing the functions of the implicative basis using a method of figurative transformations brings the task of PINF-1 and PINF-2 minimization to the level of well-researched problems within the class of disjunctive-conjunctive normal forms of Boolean functions.

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