Fixed point theorems on partial randomness

Abstract

In our former work [K. Tadaki, A statistical mechanical interpretation of algorithmic information theory, in: Local Proceedings of Computability in Europe 2008, CiE 2008, pp. 425–434, June 15–20, 2008, University of Athens, Greece. Extended Version Available at arXiv:0801.4194v1], 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∈(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).