Short lists with short programs for functions

Abstract

Let $\{\phi_p\}$ be an optimal G\"odel numbering of the family of computable functions (in Schnorr's sense), where $p$ ranges over binary strings. Assume that a list of strings $L(p)$ is computable from $p$ and for all $p$ contains a $\phi$-program for $\phi_p$ whose length is at most $\varepsilon$ bits larger that the length of the shortest $\phi$-program for $\phi_p$. We show that for infinitely many $p$ the list $L(p)$ must have $2^{|p|-\varepsilon-O(1)}$ strings. Here $\varepsilon$ is an arbitrary function of $p$.