Complete Variable-Length Codes: An Excursion into Word Edit Operations

TL;DR

This paper explores the properties of variable-length codes under specific word edit operations, analyzing their maximality and completeness in relation to binary relations like deletion, insertion, and substitution.

Contribution

It introduces a detailed study of the relationship between maximality and completeness for variable-length codes under various word edit operations.

Findings

Characterization of $ au$-independent and $ au$-closed codes

Conditions for maximality and completeness in variable-length codes

Analysis of binary relations involving deletion, insertion, substitution

Abstract

Given an alphabet A and a binary relation $\tau$ $\subseteq$ A * x A * , a language X $\subseteq$ A * is $\tau$-independent if $\tau$ (X) $\cap$ X = $\emptyset$; X is $\tau$-closed if $\tau$ (X) $\subseteq$ X. The language X is complete if any word over A is a factor of some concatenation of words in X. Given a family of languages F containing X, X is maximal in F if no other set of F can stricly contain X. A language X $\subseteq$ A * is a variable-length code if any equation among the words of X is necessarily trivial. The study discusses the relationship between maximality and completeness in the case of $\tau$-independent or $\tau$-closed variable-length codes. We focus to the binary relations by which the images of words are computed by deleting, inserting, or substituting some characters.