Devising a method of figurative transformations for minimizing boolean functions in the implicative basis

Mykhailo Solomko,Iuliia Batyshkina,Stepaniia Babych,Liudmyla Zubyk,Ihor Voitovych,Kateryna Muzychuk

doi:10.15587/1729-4061.2020.220094

Abstract

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

Highlights

The technology of designing Boolean functions included in the logical basis can resort to the implementation based on certain physical phenomena
– methods to simplify the functions of the implicative basis; – the minimization of logical schemes based on the implication functions; – the reliability of an optimal result from minimizing the implicative basis
The object of solving the task of Boolean functions sim- protocols for simplifying the functions of the implicative baplification in the implicative basis by the method of figura- sis, followed by the creation of a library of rules for the algebra tive transformations is the binary structures with repetition, of logic that have an illustration of the corresponding figuwhich are the truth tables of the assigned functions

Summary

Introduction

The technology of designing Boolean functions included in the logical basis can resort to the implementation based on certain physical phenomena. The functional completeness of the switching function system ensures the possibility to represent an arbitrary functional dependence on the assigned number of arguments by using the minimum number of basic functions (operations) These functions (operations) collectively have the property of functional completeness, and, possess the ability to synthesize a combination scheme that reproduces the functional dependence by employing a minimum number of the types of logical elements. Using the elemental basis of only one functionally complete switching function system does not ensure, in a general case, deriving an optimal combination scheme [2]. The process of minimizing logical functions occupies an important position within the technology of designing digital components In this regard, it is still an actual task to ensure the adequate compliance of a developed product with the specified cost specifications, the simplification and the warranty of obtaining the optimal result from minimizing different representations of logical functions. – methods to simplify the functions of the implicative basis; – the minimization of logical schemes based on the implication functions; – the reliability of an optimal result from minimizing the implicative basis

Literature review and problem statement

Perfect implicative normal forms of Boolean functions

Axioms and transformations in the implicative basis

Findings

10 Super-gluing the variables 11 Gluing the variables