Analytical Complexity and Signal Coding

There are two ways to describe a geometric object \(L\): the object as an image of a mapping and the object as a preimage. Every method has its own advantages and shortcomings; together, they give a complete picture. In order to compare these descriptions by complexity, one can use Kolmogorov’s approach: i.e., after the clarification of the system of basic operations, the complexity of a description is the minimum length of the defining text. Accordingly, we obtain two Kolmogorov complexities: in the first case, \(K^{+}(L)\), and in the other, \(K^{-}(L)\). Let \(Cl^n\) be the class of functions of two variables that can be represented by analytic functions of one variable and by the addition of the depth not exceeding \(n\), and let \(K^{+}(Cl^n)\) and \(K^{-}(Cl^n)\) be their corresponding Kolmogorov complexities. There are arguments in favor of the fact that, for \(n \geq 2\), the value of \(K^{-}(Cl^n)\) is very large, and the task of constructing a description of \(Cl^n\) in the form of a preimage (by defining relations) even for \(n=2\) is computationally unrealizable. Based on this observation, a signal encoding-decoding scheme is proposed, and arguments are given in favor of the fact that the decoding of a signal encoded using such a scheme is inaccessible to a quantum computer.

DOI 10.1134/S106192084010035