Fixed Point Theorems on Partial Randomness

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, such as free energy F(T), energy E(T), and statistical mechanical entropy S(T), into the theory. 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).