Walking on Words

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.