Fixed point theorems on partial randomness

TL;DR

This paper extends the statistical mechanical interpretation of algorithmic information theory by establishing fixed point theorems on partial randomness based on the computability of thermodynamic quantities.

Contribution

It proves that the computability of thermodynamic quantities like free energy, energy, and entropy yields fixed points on partial randomness, broadening previous results.

Findings

Computability of thermodynamic quantities implies fixed points on partial randomness.

Different fixed points arise from the computability of free energy versus partition function.

The results deepen the connection between computability and randomness in algorithmic information theory.

Abstract

In our former work [K. Tadaki, Local Proceedings of CiE 2008, pp.425-434, 2008], we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T, such as free energy F(T), energy E(T), and statistical mechanical entropy S(T), into the theory. These quantities are real functions of real argument T>0. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity. Namely, the computability of the value of partition function Z(T) gives a sufficient condition for T in (0,1) to be a fixed point on partial randomness. In this paper, we show that the computability of each of all the thermodynamic quantities above gives the sufficient condition also. Moreover, we show that the computability of F(T) gives completely different fixed points from the computability of Z(T).