The Tsallis entropy and the Shannon entropy of a universal probability

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.