# Descriptive Complexity of Computable Sequences Revisited

## TL;DR

This paper investigates the descriptive complexity of computable sequences, establishing bounds and relationships between different complexity measures, and clarifying open questions from prior research in the field.

## Contribution

It proves that the minimal oracle-based program length is bounded by a function of the liminf of prefix complexities, and shows this liminf is not computably bounded by the oracle program length.

## Key findings

- $C^{0'}(eta) \\le M_{\\infty}(eta)+O(1)$ for any sequence \\beta
- $M_{\\infty}(eta)$ is not bounded by any computable function of $C^{0'}(eta)$
- Results answer open questions from previous work on descriptive complexity of sequences.

## Abstract

The purpose of this paper is to answer two questions left open in [B. Durand, A. Shen, and N. Vereshchagin, Descriptive Complexity of Computable Sequences, Theoretical Computer Science 171 (2001), pp. 47--58]. Namely, we consider the following two complexities of an infinite computable 0-1-sequence $\alpha$: $C^{0'}(\alpha )$, defined as the minimal length of a program with oracle $0'$ that prints $\alpha$, and $M_{\infty}(\alpha)$, defined as $\liminf C(\alpha_{1:n}|n)$, where $\alpha_{1:n}$ denotes the length-$n$ prefix of $\alpha$ and $C(x|y)$ stands for conditional Kolmogorov complexity. We show that $C^{0'}(\alpha )\le M_{\infty}(\alpha)+O(1)$ and $M_{\infty}(\alpha)$ is not bounded by any computable function of $C^{0'}(\alpha )$, even on the domain of computable sequences.