Mechanisms of Temporal Quasi - Fractal Logic in Smart Systems

Abstract

One of the goals of digitalization is to simplify the implementation of various procedures necessary in many areas of activity, which, as a rule, require large different costs: financial, time, resource, etc. In this paper, we consider approaches to describe temporal logic from the point of view of algebra to clear up (to measure) the truth value of closed formulas of the first order language of the finite signature with equality and to clear up connections between various kinds of temporal logics. The paper is applied in nature, and one can use results received in it to construct control and planning models of smart systems. The purpose of this paper is to 1) to define the concept of a quasi-fractal algebraic system with any infinite number of levels, 2) formulate the theorem about the representation tense logic in the form of Boolean algebra with operators, and 3) use the approach to temporal logic (first order logic) from the point of view of the Category Theory [Saunders Mac Lane, Categories for the Working Mathematician, Springer (Graduate Texts in Mathematics) ISBN 0-387-98403-8] to define using this approach the notion of P – purities in the class of finite signature algebras with finite-place operations and after that in categories and in first order logics which helps one to distinguish sustainable logical systems in the sense of truth values. The paper deals with issues related to quasi - fractal temporal logic. A new operator QF is introduced for quasi - fractal algebraic systems of any infinite ordinal number of levels. So, to describe temporal logic from the point of view of algebra to clear up (to measure) the truth value of closed formulas of the first order language of the finite signature with equality and to clear up connections between various kinds of temporal logics while considering temporal logic we can define quasi - fractal time scale, or simply quasi - fractal time. Quasi - fractal temporal logic can be considered as complex system with binary connections defined by logics values - true and false, or as more complicated connections which are defined by digitalization functions. Quasi - fractal construction and fixed - point theorem helps one to highlight sustainable parts of temporal logic.