Fixed point theorems on partial randomness

In [19] we developed a statistical mechanical interpretation of AIT. In the development we introduced especially the notion of thermodynamic quantities, such as partition function Z(T ), free energy F (T ), energy E(T ), statistical mechanical entropy S(T ), and specific heat C(T ), into AIT. These quantities are real numbers which depend on temperature T , any positive real number. We then proved that if the temperature T is a computable real number with 0 < T < 1 then, for each of these thermodynamic quantities, the partial randomness of its value equals to T , where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Thus, the temperature T plays a role as the partial randomness of all the thermodynamic quantities in the statistical mechanical interpretation of AIT. In [19] we further showed that the temperature T plays a role as the partial randomness of the temperature T itself, which is a thermodynamic quantity of itself. Namely, we proved the fixed point theorem on partial randomness, 1 which states that, for every T ∈ (0, 1), if the value of partition function Z(T ) at temperature T is a computable real number, then the partial randomness of T equals to T , and therefore the compression rate of T equals to T , i.e., lim n→∞ H(T n )/n = T , where T n is the first n bits of the base-two expansion of T .

In this paper, we show that a fixed point theorem of the same form as for Z(T ) holds also for each of free energy F (T ), energy E(T ), and statistical mechanical entropy S(T ). Moreover, based on the statistical mechanical relation F (T ) = -T log 2 Z(T ), we show that the computability of F (T ) gives completely different fixed points from the computability of Z(T ).

The paper is organized as follows. We begin in Section 2 with some preliminaries to AIT and partial randomness. In Section 3, we review the previous results [19] on the statistical mechanical interpretation of AIT and the fixed point theorem by Z(T ), which is given as Theorem 3.4 in the present paper. Our main results; the fixed point theorems by F (T ), E(T ), and S(T ), are presented in Section 4, and their proofs are completed in Section 5. In the last section, we investigate some properties of the sufficient conditions for T to be a fixed point in the fixed point theorems.

We start with some notation about numbers and strings which will be used in this paper. N = {0, 1, 2, 3, . . . } is the set of natural numbers, and N + is the set of positive integers. Q is the set of rational numbers, and R is the set of real numbers. Let f : S → R with S ⊂ R. We say that f is increasing (resp., decreasing) if f (x) < f (y) (resp., f (x) > f (y)) for all x, y ∈ S with x < y. We denote by f ′ the derived function of f . Normally, o(n) denotes any function f : N + → R such that lim n→∞ f (n)/n = 0. On the other hand, O(1) denotes any function g :

{0, 1} * = {λ, 0, 1, 00, 01, 10, 11, 000, . . . } is the set of finite binary strings, where λ denotes the empty string. For any s ∈ {0, 1} * , |s| is the length of s. A subset S of {0, 1} * is called prefix-free if no string in S is a prefix of another string in S. For any partial function f , the domain of definition of f is denoted by dom f . We write “r.e.” instead of “recursively enumerable.”

Let α be an arbitrary real number. ⌊α⌋ is the greatest integer less than or equal to α, and ⌈α⌉ is the smallest integer greater than or equal to α. For any n ∈ N + , we denote by α n ∈ {0, 1} * the first n bits of the base-two expansion of α -⌊α⌋ with infinitely many zeros. For example, in the case of α = 5/8, α 6 = 101000.

We say that a real number α is computable if there exists a total recursive function f :

n for all n ∈ N + . We say that α is left-computable if there exists a total recursive function g : N + → Q such that g(n) ≤ α for all n ∈ N + and lim n→∞ g(n) = α. On the other hand, we say that a real number α is right-computable if -α is left-computable. The following (i) and (ii) then hold:

(i) A real number α is computable if and only if α is both left-computable and right-computable.

See e.g. Weihrauch [22] for the detail of the treatment of the computability of real numbers.

In the following we concisely review some definitions and results of algorithmic

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