On Stricter Reachable Repetitiveness Measures*

TL;DR

This paper introduces NU-systems, combining morphism compositions with macro schemes, revealing that the most general repetitiveness measure can be even smaller than previously believed, thus advancing understanding of sequence compressibility.

Contribution

It defines NU-systems as a new mechanism for characterizing repetitive sequences and shows their size can be smaller than existing measures like  and , indicating greater potential for sequence compression.

Findings

NU-systems can have size o() for some strings

NU-systems size  is reachable and smaller than previous bounds

Repetitiveness measures can be even more restrictive than known measures

Abstract

The size $b$ of the smallest bidirectional macro scheme, which is arguably the most general copy-paste scheme to generate a given sequence, is considered to be the strictest reachable measure of repetitiveness. It is strictly lower-bounded by measures like $\gamma$ and $\delta$, which are known or believed to be unreachable and to capture the entropy of repetitiveness. In this paper we study another sequence generation mechanism, namely compositions of a morphism. We show that these form another plausible mechanism to characterize repetitive sequences and define NU-systems, which combine such a mechanism with macro schemes. We show that the size $\nu \leq b$ of the smallest NU-system is reachable and can be $o(\delta)$ for some string families, thereby implying that the limit of compressibility of repetitive sequences can be even smaller than previously thought. We also derive several other results characterizing $\nu$.