基于可压缩弦的Chaitin Ω数的一种新表示
A new representation of Chaitin Ωnumber based on compressible strings
摘要：
柴廷在1975年介绍了他的Ω数作为一个具体的例子随机实。实Ω是基于最优无前缀机器U的所有停机输入的集合来定义的，这是一种用于定义程序大小复杂度概念的通用解码算法。Chaitin发现Ω是随机的，因为Ω的二进制扩展的前n位解决了U对于所有长度不超过n的二进制输入的停机问题。本文引入了Chaitin Ω数的一种新表示Θ。实数Θ是基于所有可压缩字符串的集合定义的。我们研究了Θ的性质，证明了Θ是随机的。此外，我们将Θ推广到两个方向Θ（T）和\bar Θ（T），其中实T>0.然后我们研究它们的性质。特别地，我们证明了实Θ（T）的可计算性给出了（0，1）中的实T是部分随机性上的不动点的充分条件，即，以满足T的压缩率等于T的条件。
英文摘要：
In 1975 Chaitin introduced his Ωnumber as a concrete example of random real. The real Ωis defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ωto be random by discovering the property that the first n bits of the base-two expansion of Ωsolve the halting problem of U for all binary inputs of length at most n. In this paper, we introduce a new representation Θof Chaitin Ωnumber. The real Θis defined based on the set of all compressible strings. We investigate the properties of Θand show that Θis random. In addition, we generalize Θto two directions Θ(T) and \barΘ(T) with a real T>0. We then study their properties. In particular, we show that the computability of the real Θ(T) gives a sufficient condition for a real T in (0,1) to be a fixed point on partial randomness, i.e., to satisfy the condition that the compression rate of T equals to T.
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