Kolmogorov's legacy: Algorithmic Theory of Informatics and Kolmogorov Programmable Technology

Abstract: In this survey, we explore Andrei Nikolayevich Kolmogorov's seminal work in just one of his many facets: its influence Computer Science especially his viewpoint of what herein we call 'Algorithmic Theory of Informatics.' Can a computer file 'reduce' its 'size' if we add to it new symbols? Do equations of state like second Newton law in Physics exist in Computer Science? Can Leibniz' principle of identification by indistinguishability be formalized? In the computer, there are no coordinates, no distances, and no dimensions; most of traditional mathematical approaches do not work. The computer processes finite binary sequences i.e. the sequences of 0 and 1. A natural question arises: Should we continue today, as we have done for many years, to approach Computer Science problems by using classical mathematical apparatus such as 'mathematical modeling'? The first who drew attention to this question and gave insightful answers to it was Kolmogorov in 1960s. Kolmogorov's empirical postulate about existence of a program that translates 'a natural number into its binary record and the record into the number' formulated in 1958 represents a hint of Kolmogorov's approach to Computer Science. Following his ideas, we interpret Kolmogorov algorithm, Kolmogorov machine, and Kolmogorov complexity in the context of modern information technologies showing that they essentially represent fundamental elements of Algorithmic Theory of Informatics, Kolmogorov Programmable Technology, and new Komputer Mathematics i.e. Mathematics of computers.