Isometry groups of formal languages for generalized Levenshtein distances

TL;DR

This paper investigates the structure of isometry groups of formal languages under generalized Levenshtein distances, establishing bounds on word length differences and embedding properties of these groups.

Contribution

It proves bounds on length differences under isometries for certain Levenshtein distances and shows that these groups embed into an infinite product of symmetric groups.

Findings

Bound on length difference under isometries depends only on the language

Isometry groups embed into an infinite product of symmetric groups

Constructed examples demonstrate the bounds are essentially tight

Abstract

This article is a partial answer to the question of which groups can be represented as isometry groups of formal languages for generalized Levenshtein distances. Namely, it is proved that for any language the modulus of the difference between the lengths of its words and the lengths of their images under isometry for an arbitrary generalized Levenshtein distance that satisfies the condition that the weight of the replacement operation is less than twice the weight of the removal operation is bounded above by a constant that depends only on the language itself. From this, in particular, it follows that the isometry groups of formal languages with respect to such metrics always embed into the group $\Pi_{n=1}^\infty S_{n}$. We also construct a number of examples showing that this estimate is, in a certain sense, unimprovable.