指数串及其编辑距离
Exponent-Strings and Their Edit Distance
AI总结：
本文提出指数串以扩展传统串支持实数指数，基于半群理论证明其代数性质，并开发了exp-edit distance算法来度量兼具离散与连续性质的$\mathbb{R}^+$-指数串相似度。
摘要：
指数串是传统串的扩展，可包含实数指数以表示字符数量。这种新表示克服了传统离散串的局限性，能够为包含声音时长的语音转录等应用提供精确的数据表示。尽管指数串的应用侧重于实数指数，其形式化定义使用任意半群。对于任意半群$S$，$S$-指数串允许使用$S$的元素作为指数。我们研究了$S$-指数串的代数性质，并进一步证明$\mathbb{R}^+$-指数串是串的自然扩展。受计算口语语音序列与正确语音序列之间相似度问题的启发，我们开发了exp-edit distance——一种专门设计用于度量$\mathbb{R}^+$-指数串之间相似度的指标。通过扩展传统串编辑距离以处理连续值，exp-edit distance能够处理兼具离散和连续性质的$\mathbb{R}^+$-指数串。我们的探索包括exp-edit distance的严格数学公式化及其计算算法。
英文摘要：
An exponent-string is an extension of traditional strings that can incorporate real-numbered exponents, indicating the quantity of characters. This novel representation overcomes the limitations of traditional discrete string by enabling precise data representation for applications such as phonetic transcription that contains sound duration. Although applications of exponent-string are focused on exponent-string with real-numbered exponents, formal definition uses arbitrary semigroup. For any semigroup $S$, $S$-exponent-strings are allowed to have elements of $S$ as exponents. We investigate algebraic properties of $S$-exponent-strings and further justify $\mathbb{R}^+$-exponent-string is a natural extension of the string. Motivated by the problem of calculating the similarity between spoken phone sequence and correct phone sequence, we develop exp-edit distance -- a specialized metric designed to measure the similarity between $\mathbb{R}^+$-exponent-strings. By extending the traditional string edit distance to handle continuous values, exp-edit distance deals with $\mathbb{R}^+$-exponent-strings that embody both discrete and continuous properties. Our exploration includes a rigorous mathematical formulation of exp-edit distance and an algorithm to compute it.
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。
