We study edit distances between strings, based on weighted operations such as character substitutions, insertions, deletions, and consolidations and fragmentations. The two latter operations transform a sequence of characters into one character and vice-versa. They correspond to the compression and expansion in Dynamic Time-Warping algorithms for speech recognition and are used for the formal analysis of written music.

We show that such edit distances are not computable in general, and propose weighted automata constructions to compute restricted cases of edit distances, taking into account both consolidations and deletions, or both fragmentations and insertions. Assuming that the operation ruleset has a constant size, these constructions are polynomial into the lengths of the involved strings. We finally show that the optimal weight of sequences made of consolidations chained with fragmentations, in that order, is computable for arbitrary rulesets, and not computable for some rulesets when reversing the order of fragmentations and consolidations.

我们研究字符串之间的编辑距离，基于加权操作，例如字符替换、插入、删除以及合并和碎片。后两个操作将一系列字符转换为一个字符，反之亦然。它们对应于用于语音识别的动态时间扭曲算法中的压缩和扩展，并用于书面音乐的形式分析。

我们表明这种编辑距离一般是不可计算的，并提出加权自动机构造来计算编辑距离的限制情况，同时考虑合并和删除，或碎片和插入。假设操作规则集具有恒定大小，这些构造是涉及字符串长度的多项式。我们最终表明，按该顺序由与碎片链接的合并组成的序列的最佳权重对于任意规则集是可计算的，并且在颠倒碎片和合并的顺序时对于某些规则集是不可计算的。