Walking on Words

TL;DR

Walking on Words investigates how a word $u$ can generate another word $w$ through constrained walks on $u$, with the central notion of a primitive generator. The authors prove that every word has a primitive generator, unique up to reversal, and that equal outputs $u^f$ and $u^g$ from distinct walks are fully characterized by palindromic structure via the defect relation $\Delta(u)$ and its transitive closure $\Delta^*(u)$. They further separate the theory by showing that primitive generators are unique up to reversal, while the corresponding generating walks may not be, and provide a complete framework using hesitations, vacillations, and reflections to describe when different walks yield the same result. The paper applies these ideas to morphic words, notably showing that in the $k$-bonacci family from some index onward all terms share the same primitive generator, illustrating the interaction between palindromic richness and generating paths. Overall, the work advances string combinatorics by linking primitive generators, palindromic defects, and morphic word structures, with implications for logic and formal languages.

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.

Figures (14)

• Figure 1: Generation and minimal legs.
• Figure 2: The walk $g$ has waypoints at $Y=V$ and at $Z$.
• Figure 3: The walk $g$ has a waypoint at $Y$ with $k = |V-Y| \geq 1$.
• Figure 4: The location of $Z$ with respect to $V{+} \frac{3}{2}k$ in Case (iii).
• Figure 5: The walk $g$ has a waypoint at $Y$.

Theorems & Definitions (27)

• Theorem 1
• Corollary 2
• proof
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof
• proof
• proof