Robustness of statistical mechanical interpretation of algorithmic information theory

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed in our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425-434, 2008], where we introduced the thermodynamic quantities into AIT. In this paper, we reveal a certain sort of the robustness of statistical mechanical interpretation of AIT. The thermodynamic quantities in AIT are originally defined based on the set of all programs, i.e., all halting inputs, for an optimal prefix-free machine, which is a universal decoding algorithm used to define the notion of program-size complexity. We show that we can recover the original properties of the thermodynamic quantities in AIT if we replace all programs by all minimal-size programs in the definitions of the thermodynamic quantities in AIT. The results of this paper illustrate the generality and validity of the statistical mechanical interpretation of AIT.