通过前缀-后缀复制操作生成斐波那契单词
Generating Fibonacci Words via the Prefix--Suffix Duplication Operation
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中文总结 AI 辅助
研究通过前缀-后缀复制操作生成斐波那契单词，解决并强化杜米特兰猜想，证明仅用界限为3的前缀复制可从特定斐波那契单词生成其他单词，且界限3最优，还给出线性时间算法。
中文摘要 AI 辅助
有限和无限的斐波那契单词是词组合学中的经典对象。受生物启发的语言操作是研究有限和无限单词如何通过局部重写机制产生的有用工具。例如，后缀平方完成操作可生成无限斐波那契单词以及其他无限单词。前缀-后缀复制操作产生一种单词语言。杜米特兰猜想具有相同索引奇偶性的斐波那契单词可通过此操作的有界复制长度变体相互生成。本文解决并强化了该猜想，表明对于所有1≤p≤n，仅使用界限k≥3的前缀复制就能从F₂ₚ生成F₂ₙ，从F₂ₚ₊₁生成F₂ₙ₊₁，还证明界限3是最优的，并给出一个算法，该算法能在目标斐波那契单词长度的线性时间内产生见证推导。
英文摘要
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 $1 \leq p \leq n$, 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 $k \geq 3$. 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.
