[论文解读] Primitive words and roots of words

Gerhard Lischke|arXiv (Cornell University)|Apr 22, 2011

semigroups and automata theory参考文献 21被引用 9

一句话总结

本文全面综述了形式语言理论中原始词及其根的概念，重点探讨其组合性质、与乔姆斯基层级的关系以及计算复杂度。文章引入并表征了六种不同的根函数（根、sroot、hroot、ssroot、shroot、hhroot），证明了洛曼词的存在性——即所有六种根均不同的词，并建立了其构造条件，从而解决了2010年提出的开放问题。

ABSTRACT

In the algebraic theory of codes and formal languages, the set $Q$ of all primitive words over some alphabet $\zi $ has received special interest. With this survey article we give an overview about relevant research to this topic during the last twenty years including own investigations and some new results. In Section 1 after recalling the most important notions from formal language theory we illustrate the connection between coding theory and primitive words by some facts. We define primitive words as words having only a trivial representation as the power of another word. Nonprimitive words (without the empty word) are exactly the periodic words. Every nonempty word is a power of an uniquely determined primitive word which is called the root of the former one. The set of all roots of nonempty words of a language is called the root of the language. The primitive words have interesting combinatorial properties which we consider in Section 2. In Section 3 we investigate the relationship between the set $Q$ of all primitive words over some fixed alphabet and the language classes of the Chomsky Hierarchy and the contextual languages over the same alphabet. The computational complexity of the set $Q$ and of the roots of languages are considered in Section 4. The set of all powers of the same degree of all words from a language is the power of this language. We examine the powers of languages for different sets of exponents, and especially their regularity and context-freeness, in Section 5, and the decidability of appropriate questions in Section 6. Section 7 is dedicated to several generalizations of the notions of periodicity and primitivity of words.

研究动机与目标

• 综述过去二十年间关于原始词及其根的研究成果。
• 研究原始词在乔姆斯基层级和上下文相关语言类中的结构与计算性质。
• 分析原始词集合及语言根的计算复杂度。
• 推广周期性与原始性的概念，为词定义六种不同的根函数。
• 解决所有六种根函数结果互不相同的词的存在性问题，引入洛曼词的概念。

提出的方法

• 将原始词定义为其他词的幂次，将周期词定义为非原始的非空词。
• 基于前缀与后缀关系，引入六种根函数：根、sroot、hroot、ssroot、shroot、hhroot。
• 利用字典序最小性，识别出k=1至6的最小k-根词。
• 应用组合引理（如引理58、59、60）刻画非原始词及其在扩展下的封闭性。
• 通过充分条件构造洛曼词：u = w^{k1}vw^{k2}vw^{k1}vw^{k3}vw^{k3−k1}，其中2 ≤ k1 < k2 < k3 ≤ 2k1。
• 证明广义原始词集合的非上下文无关性及非上下文相关语言性质。

实验结果

研究问题

• RQ1 是否存在某些词，使得所有六种根函数（根、sroot、hroot、ssroot、shroot、hhroot）的结果均不相同？
• RQ2 对于每个k从1到6，字典序最小的k-根词是什么？
• RQ3 是否存在强k-根词（周期k-根词）满足k=5和k=6？
• RQ4 构造洛曼词的充分条件是否也是必要条件？
• RQ5 广义原始词集合（如SQ、HQ、SSQ）与上下文无关语言类及上下文相关语言类有何关系？

主要发现

• 字典序最小的6-根词（洛曼词）为：ababaabababaababaababababaabab。
• 不存在字典序最小的强6-根词，此结论由定理56证明。
• 首个已知的洛曼词由乔治·洛曼于2010年发现，证实了存在六种不同根的词。
• 给出了构造洛曼词的充分条件：u = w^{k1}vw^{k2}vw^{k1}vw^{k3}vw^{k3−k1}，其中2 ≤ k1 < k2 < k3 ≤ 2k1。
• 集合SQ、HQ、SSQ、SHQ、HHQ均为非上下文无关语言，且不属于任何类型的上下文相关语言。
• 识别非原始词的时间复杂度上界为n²，该界对SQ是最优的，可能对其他广义集合也最优，但部分情况仍为开放问题。