Automatic complexity of Fibonacci and Tribonacci words

TL;DR

This paper investigates the automatic complexity rates of infinite Fibonacci and Tribonacci words, demonstrating the existence of words with intermediate nondeterministic automatic complexity rates between 0 and 1/2.

Contribution

It proves the existence of infinite Fibonacci and Tribonacci words with intermediate automatic complexity rates, a novel result in complexity theory of infinite words.

Findings

Fibonacci and Tribonacci words have intermediate automatic complexity rates.

Intermediate rates are strictly between 0 and 1/2.

Existence of such words was previously unknown.

Abstract

For a complexity function $C$, the lower and upper $C$-complexity rates of an infinite word $\mathbf{x}$ are \[ \underline{C}(\mathbf x)=\liminf_{n\to\infty} \frac{C(\mathbf{x}\upharpoonright n)}n,\quad \overline{C}(\mathbf x)=\limsup_{n\to\infty} \frac{C(\mathbf{x}\upharpoonright n)}n \] respectively. Here $\mathbf{x}\upharpoonright n$ is the prefix of $x$ of length $n$. We consider the case $C=\mathrm{A_N}$, the nondeterministic automatic complexity. If these rates are strictly between 0 and $1/2$, we call them intermediate. Our main result is that words having intermediate $\mathrm{A_N}$-rates exist, viz. the infinite Fibonacci and Tribonacci words.