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.