On Generating Binary Words Palindromically

TL;DR

This paper investigates how finite words, especially binary words, can be generated by minimal sets of palindromic relations, revealing that some infinite words require at least three such relations for some factors, while others have bounded or unbounded functions.

Contribution

It introduces the function μ(u) measuring the minimal palindromic relations needed to generate words and classifies binary words based on this measure, including Sturmian and Thue-Morse words.

Findings

Every aperiodic infinite word contains a factor with μ(u) ≥ 3.

Some infinite words have μ(u) ≤ 3 for all factors.

The function μ is unbounded for the Thue-Morse word.

Abstract

We regard a finite word $u=u_1u_2\cdots u_n$ up to word isomorphism as an equivalence relation on $\{1,2,\ldots, n\}$ where $i$ is equivalent to $j$ if and only if $x_i=x_j.$ Some finite words (in particular all binary words) are generated by "{\it palindromic}" relations of the form $k\sim j+i-k$ for some choice of $1\leq i\leq j\leq n$ and $k\in \{i,i+1,\ldots,j\}.$ That is to say, some finite words $u$ are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function $\mu(u)$ defined as the least number of palindromic relations required to generate $u.$ We show that every aperiodic infinite word must contain a factor $u$ with $\mu(u)\geq 3,$ and that some infinite words $x$ have the property that $\mu(u)\leq 3$ for each factor $u$ of $x.$ We obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast for the Thue-Morse word, we show that the function $\mu$ is unbounded.