The sum $2^{\mathit{KA}(x)-\mathit{KP}(x)}$ over all prefixes $x$ of some binary sequence can be infinite

Abstract

We consider two quantities that measure complexity of binary strings: $\mathit{KA}(x)$ is defined as the minus logarithm of continuous a priori probability on the binary tree, and $\mathit{KP}(x)$ denotes prefix complexity of a binary string $x$. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence $\omega$ such that the sum of $2^{\mathit{KA}(x)-\mathit{KP}(x)}$ over all prefixes $x$ of $\omega$ is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.