Consider a subset of positive integers $S$. In this paper, we reduce the upper bound on the length of a minimum program that enumerates $S$ in terms of the probability of $S$ being enumerated by a random program. So far, the best-known upper bound was given by Solovay. Solovay proved that the minimum length of a program enumerating $S$ is bounded by $3$ times minus binary logarithm of the probability that a random program enumerates $S$. Later, Vereshchagin showed that the constant can be improved from $3$ to $2$ for finite sets. By improving the method proposed by Solovay, we demonstrate that any bound for finite sets implies the same bound for infinite sets, modulo logarithmic factors. Thus, the constant can be replaced by $2$ for every set $S$ due to the result of Vereshchagin.

翻译：考虑一个正整数子集 $S$。本文中，我们降低了用随机程序枚举 $S$ 的概率来刻画枚举 $S$ 的最短程序长度的上界。目前，最佳上界由 Solovay 给出。Solovay 证明，枚举 $S$ 的最短程序长度不超过随机程序枚举 $S$ 的概率的负二进制对数的 $3$ 倍。随后，Vereshchagin 指出，对于有限集，该常数可从 $3$ 改进为 $2$。通过改进 Solovay 提出的方法，我们证明：对有限集的任何上界在忽略对数因子的情形下均可推广至无限集。因此，基于 Vereshchagin 的结果，对于任意集合 $S$，该常数可替换为 $2$。