Enumerating Complexity Revisited

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.