MATHEMATICAL DESCRIPTION OF THE DIFFERENTIATION OPERATION IN THE LOGICAL-TIME ENVIRONMENT

Abstract

Effective and timely processing of information is one of the most important problems of creating means of its processing at the level of human perception and thinking. Analytical processing of digital signals in a graphical or purely digital representation is somewhat limited and does not take into account the dynamics of signals and messages. The classical apparatus of logic is insufficient to describe the dynamics of system behavior over time. Therefore, it is important to develop models of so-called Boolean differential calculus, as this approach is based on the general concept of changing the logical variable, which will lead to a universal, in terms of dynamics, system of concepts and operations. To facilitate the preprocessing of dynamic digital variables and signals, the logic-time function of multivalued logic can be used. The purpose of this article is a mathematical representation of the differentiation of logic-time functions of multi-valued logic in the index form of the record using the simulation of its scheme. The paper shows the expediency of the idea of replacing an arbitrary digital signal (variable) that changes over time with a logic-time function, which allows to facilitate the preliminary analytical processing of digital signals and variables using the properties of such functions. The article presents a new mathematical apparatus for describing logic-time functions of multivalued logic and individual operations on them using modeling of known implementation schemes. The paper considers the peculiarities of the derivative of the multivalued logical-time function (LMF), as one of the most used and basic operations used in the study of signals and images. Its characteristics and features are shown for functions presented in index form. The general expression of the derivative of the n order is obtained and it is shown that different logic-time functions can have the same derivative. The concepts of the left and right derivative of the LMF are introduced and the relationship between them is shown. The properties of the k derivative of the BLMF were considered and the expressions for the derivative conjunction and disjunction were obtained. A possible structural scheme of the differentiator is presented, which opens up the possibility of hardware processing of multi-valued LMFs.