部分随机性的不动点定理
Fixed point theorems on partial randomness
摘要：
在我们以前的工作[K。Tadaki，Local Proceedings of CiE 2008，pp.425-434，2008]，我们通过将温度T下的热力学量的概念（例如自由能F（T）、能量E（T）和统计机械熵S（T））引入到理论中，开发了算法信息论的统计机械解释。这些量是实变元T>0的实函数。然后，我们发现，在解释中，温度T等于所有这些热力学量的值的部分随机性，其中部分随机性的概念是通过程序大小复杂性来更强地表示压缩率。此外，我们还证明了这种情况对于作为热力学量的温度本身也成立。也就是说，配分函数Z（T）值的可计算性给出了T在（0，1）中是部分随机不动点的充分条件。本文证明了上述热力学量的可计算性也给出了充分条件。此外，我们证明了F（T）的可计算性给出了与Z（T）的可计算性完全不同的不动点。
英文摘要：
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 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 in (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).
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