RePair文法是斐波那契字的最小文法
RePair Grammars are the Smallest Grammars for Fibonacci Words
摘要：
基于语法的压缩是一种无损数据压缩方案，其通过仅生成$w$的上下文无关语法来表示给定字符串$w$。虽然计算生成给定字符串$w$的最小文法通常是NP困难的，但已经提出了许多在实践中工作良好的基于多项式时间文法的压缩器。由Larsson和Moffat在1999年提出的RePair是一种基于语法的压缩器，它递归地替换字符串中最频繁出现的二元组的所有可能出现。由于可以有多种选择来替换最频繁的二元组，因此RePair的不同实现可能会导致不同的语法。在本文中，我们表明，生成斐波那契字的最小文法$F_k$可以完全由RePair表征，其中$F_k$表示第$k$斐波那契字。即，由RePair的任何实现生成的用于$F_k$的所有语法是用于$F_k$的最小语法，并且没有其他语法可以是用于$F_k$的最小语法。据我们所知，斐波那契字是第一个非平凡的无限系列的字符串，其中RePair是最佳的。
英文摘要：
Grammar-based compression is a loss-less data compression scheme that represents a given string $w$ by a context-free grammar that generates only $w$. While computing the smallest grammar which generates a given string $w$ is NP-hard in general, a number of polynomial-time grammar-based compressors which work well in practice have been proposed. RePair, proposed by Larsson and Moffat in 1999, is a grammar-based compressor which recursively replaces all possible occurrences of a most frequently occurring bigrams in the string. Since there can be multiple choices of the most frequent bigrams to replace, different implementations of RePair can result in different grammars. In this paper, we show that the smallest grammars generating the Fibonacci words $F_k$ can be completely characterized by RePair, where $F_k$ denotes the $k$-th Fibonacci word. Namely, all grammars for $F_k$ generated by any implementation of RePair are the smallest grammars for $F_k$, and no other grammars can be the smallest for $F_k$. To the best of our knowledge, Fibonacci words are the first non-trivial infinite family of strings for which RePair is optimal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
