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

Given an alphabet A and a binary relation $τ$ $\subseteq$ A * x A * , a language X $\subseteq$ A * is $τ$-independent if $τ$ (X) $\cap$ X = $\emptyset$; X is $τ$-closed if $τ$ (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 $τ$-independent or $τ$-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.