Generating Fibonacci Words via the Prefix--Suffix Duplication Operation

Abstract

The finite and infinite Fibonacci words are classical objects in combinatorics on words. Bio-inspired language operations provide a useful tool for studying how finite and infinite words can arise via local rewriting mechanisms. For example, the suffix square completion operation is known to generate the infinite Fibonacci word, as well as other infinite words such as the Thue--Morse word and the period-doubling word. The prefix--suffix duplication operation produces a language of words formed by appending prefixes or suffixes of a word $w$ to the front or back of $w$ respectively, and Dumitran conjectured that Fibonacci words of the same index parity can be generated from one another by the bounded duplication length variant of this operation. In this paper, we resolve and strengthen Dumitran's conjecture. We show that, for all $1\le p\le n$, it is possible to generate the Fibonacci word $F_{2n}$ from $F_{2p}$, and $F_{2n+1}$ from $F_{2p+1}$, using only prefix duplications with a bound of $k\ge 3$. We furthermore show that this bound of $3$ is optimal, and we give an algorithm that produces a witness derivation in time linear in the length of the target Fibonacci word.