On prefix palindromic length of automatic words

TL;DR

This paper investigates the prefix palindromic length of automatic words, proving regularity for certain classes and providing formulas for specific cases, while suggesting that some automatic words may have non-regular prefix palindromic length functions.

Contribution

It proves that the prefix palindromic length function is $k$-regular for $k$-automatic words with finitely many palindromes and derives explicit formulas for the paperfolding and Rudin-Shapiro words.

Findings

The function is $k$-regular for certain automatic words.

Explicit formulas are derived for the paperfolding and Rudin-Shapiro words.

Evidence suggests some automatic words have non-regular prefix palindromic length functions.

Abstract

The prefix palindromic length $\mathrm{PPL}_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. Since 2013, it is still unknown if $\mathrm{PPL}_{\mathbf{u}}(n)$ is unbounded for every aperiodic infinite word $\mathbf{u}$, even though this has been proven for almost all aperiodic words. At the same time, the only well-known nontrivial infinite word for which the function $\mathrm{PPL}_{\mathbf{u}}(n)$ has been precisely computed is the Thue-Morse word $\mathbf{t}$. This word is $2$-automatic and, predictably, its function $\mathrm{PPL}_{\mathbf{t}}(n)$ is $2$-regular, but is this the case for all automatic words? In this paper, we prove that this function is $k$-regular for every $k$-automatic word containing only a finite number of palindromes. For two such words, namely the paperfolding word and the Rudin-Shapiro word, we derive a formula for this function. Our computational experiments suggest that generally this is not true: for the period-doubling word, the prefix palindromic length does not look $2$-regular, and for the Fibonacci word, it does not look Fibonacci-regular. If proven, these results would give rare (if not first) examples of a natural function of an automatic word which is not regular.