Weighted Prefix Normal Words: Mind the Gap

TL;DR

This paper generalizes the concept of prefix normal words from binary to arbitrary finite alphabets, exploring their properties, expressiveness, and the existence of a normalized form.

Contribution

It introduces weighted prefix normality for finite words over arbitrary alphabets and develops a method to obtain a generalized prefix normal form.

Findings

Weighted prefix normality is more expressive than binary prefix normality.

The paper characterizes peculiarities of weighted prefix normality.

A technique for obtaining a generalized prefix normal form is presented.

Abstract

A prefix normal word is a binary word whose prefixes contain at least as many 1s as any of its factors of the same length. Introduced by Fici and Lipt\'ak in 2011 the notion of prefix normality is so far only defined for words over the binary alphabet. In this work we investigate a generalisation for finite words over arbitrary finite alphabets, namely weighted prefix normality. We prove that weighted prefix normality is more expressive than binary prefix normality. Furthermore, we investigate the existence of a weighted prefix normal form since weighted prefix normality comes with several new peculiarities that did not already occur in the binary case. We characterise these issues and finally present a standard technique to obtain a generalised prefix normal form for all words overarbitrary, finite alphabets.