Phase Transition and Strong Predictability

Abstract: 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 notion of thermodynamic quantities into AIT. These quantities are real functions of temperature T>0. The values of all the thermodynamic quantities diverge when T exceeds 1. This phenomenon corresponds to phase transition in statistical mechanics. In this paper we introduce the notion of strong predictability for an infinite binary sequence and then apply it to the partition function Z(T), which is one of the thermodynamic quantities in AIT. We then reveal a new computational aspect of the phase transition in AIT by showing the critical difference of the behavior of Z(T) between T=1 and T<1 in terms of the strong predictability for the base-two expansion of Z(T).