The Tsallis entropy and the Shannon entropy of a universal probability

TL;DR

This paper explores the properties of Tsallis and Shannon entropies in the context of algorithmic randomness, analyzing their convergence and relation to program-size complexity of universal probabilities.

Contribution

It investigates the convergence and randomness degree of Tsallis and Shannon entropies for universal probabilities using algorithmic information theory.

Findings

Determines conditions for convergence or divergence of entropies.

Evaluates the degree of randomness when entropies converge.

Links entropy properties to program-size complexity in universal probabilities.

Abstract

We study the properties of Tsallis entropy and Shannon entropy from the point of view of algorithmic randomness. In algorithmic information theory, there are two equivalent ways to define the program-size complexity K(s) of a given finite binary string s. In the standard way, K(s) is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output s. In the other way, the so-called universal probability m is introduced first, and then K(s) is defined as -log_2 m(s) without reference to the concept of program-size. In this paper, we investigate the properties of the Shannon entropy, the power sum, and the Tsallis entropy of a universal probability by means of the notion of program-size complexity. We determine the convergence or divergence of each of these three quantities, and evaluate its degree of randomness if it converges.