Some Remarks on Marginal Code Languages

TL;DR

The paper develops two uniform frameworks—partial orders and transducers—to define and study marginal independent languages, namely $k$-$\alpha$ and finitely-$\alpha$ generalizations of classical code concepts (prefix/suffix/infix). It establishes that every $k$-$\alpha$ language forms a $(k+3)$-independent property with maximal extensions, while finitely-$\alpha$ properties need not be Jürgensen properties, demonstrated via a right-infinite partial order result. It further analyzes decision problems: the fixed-$k$ marginal satisfaction problems are PSPACE-complete (and PSPACE-hard for $k$-infix), with recent work showing uniform satisfaction and maximality can be decided when the property is described by a transducer, and that finitely marginal versions admit efficient solutions (e.g., $O(|\bm{a}|^2)$). Collectively, the work delineates the computational boundaries for marginal code properties and provides unified methodologies to extend existing code theories to marginal variants, with implications for uniform reasoning about language independence and maximality.

Abstract

A prefix code L satisfies the condition that no word of L is a proper prefix of another word of L. Recently, Ko, Han and Salomaa relaxed this condition by allowing a word of L to be a proper prefix of at most k words of L, for some `margin' k, introducing thus the class of k-prefix-free languages, as well as the similar classes of k-suffix-free and k-infix-free languages. Here we unify the definitions of these three classes of languages into one uniform definition in two ways: via the method of partial orders and via the method of transducers. Thus, for any known class of code-related languages definable via the transducer method, one gets a marginal version of that class. Building on the techniques of Ko, Han and Salomaa, we discuss the \emph{uniform} satisfaction and maximality problems for marginal classes of languages.