Walking on Words

TL;DR

This paper investigates the process of generating words through a walk on the input, revealing that most words have exactly two primitive generators, which are minimal generating sequences.

Contribution

It introduces the concept of primitive generators in the context of walks on words and proves that, barring degenerate cases, each word has exactly two such generators.

Findings

Most words have exactly two primitive generators.

The concept of primitive generators is formally defined and characterized.

The results hold except for some degenerate cases.

Abstract

Take any word over some alphabet. If it is non-empty, go to any position and print out the letter being scanned. Now repeat the following any number of times (possibly zero): either stay at the current letter, or move one letter leftwards (if possible) or move one letter rightwards (if possible); then print out the letter being scanned. In effect, we are going for a walk on the input word. Let u be the infix of the input word comprising the visited positions, and w the word printed out (empty if the input word is). Since any unvisited prefix or suffix of the input word cannot influence w, we may as well discard them, and say that u generates w. We ask: given a word w, what words u generate it? The answer is surprising. Call u a primitive generator of w if u generates w and is not generated by any word shorter than u. We show that, excepting some degenerate cases, every word has precisely two primitive generators.