We reduce the best-known upper bound on the length of a program that enumerates a set in terms of the probability of it being enumerated by a random program. We prove a general result that any linear upper bound for finite sets implies the same linear bound for infinite sets. So far, the best-known upper bound was given by Solovay. He showed that the minimum length of a program enumerating a subset $S$ of natural numbers is bounded by minus three binary logarithms of the probability that a random program will enumerate $S$. Later, Vereshchagin showed that the constant can be improved from three to two for finite sets. In this work, using an improvement of the method proposed by Solovay, we demonstrate that any bound for finite sets implies the same for infinite sets, modulo logarithmic factors. Using Vereshchagin's result, we improve the current best-known upper bound from three to two.

翻译：我们降低了描述一个集合的程序长度的最佳已知上界，该上界基于随机程序枚举该集合的概率。我们证明了一个一般性结果：对于有限集的任何线性上界，同样适用于无限集。迄今为止，最佳已知上界由Solovay给出。他证明，枚举自然数子集$S$的程序的最小长度，受限于随机程序枚举$S$的概率的负三倍二进制对数。随后，Vereshchagin表明，对于有限集，该常数可从三改进为二。在本工作中，通过改进Solovay提出的方法，我们证明：在对数因子范围内，任何有限集的上界同样适用于无限集。利用Vereshchagin的结果，我们将当前最佳已知上界从三改进为二。