Fixed point theorems on partial randomness

In our former work [K. Tadaki, Local Proceedings of CiE 2008,pp. 425---434, 2008], we developed a statistical mechanicalinterpretation of algorithmic information theory by introducing thenotion of thermodynamic quantities, such as free energyF (T ), energy E (T ), andstatistical mechanical entropy S (T ), into thetheory. We then discovered that, in the interpretation, thetemperature T equals to the partial randomness of thevalues of all these thermodynamic quantities, where the notion ofpartial randomness is a stronger representation of the compressionrate by program-size complexity. Furthermore, we showed that thissituation holds for the temperature itself as a thermodynamicquantity. Namely, the computability of the value of partitionfunction Z (T ) gives a sufficient condition forT ⊆ (0,1) to be a fixed point on partial randomness.In this paper, we show that the computability of each of all thethermodynamic quantities above gives the sufficient condition also.Moreover, we show that the computability of F (T )gives completely different fixed points from the computability ofZ (T ).

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