Normal Sequences with Non-Maximal Automatic Complexity

TL;DR

This paper explores the automatic complexity of normal sequences, showing that some can have minimal complexity rates while still being normal, and provides bounds for specific sequences like Champernowne.

Contribution

It demonstrates the existence of normal sequences with minimal automatic complexity rates and establishes upper bounds for the complexity of Champernowne sequences.

Findings

Existence of normal sequences with $I(T)=0$ and $S(T) \\leq 1/2$

Champernowne sequence has $S(C) \\leq 2/3$

Insights into automatic complexity of normal sequences

Abstract

This paper examines Automatic Complexity, a complexity notion introduced by Shallit and Wang in 2001. We demonstrate that there exists a normal sequence $T$ such that $I(T) = 0$ and $S(T) \leq 1/2$, where $I(T)$ and $S(T)$ are the lower and upper automatic complexity rates of $T$ respectively. We furthermore show that there exists a Champernowne sequence $C$, i.e. a sequence formed by concatenating all strings of length $1$ followed by concatenating all strings of length $2$ and so on, such that $S(C) \leq 2/3$.