Walking on Words

Any function f with domain {1, … , m} and co-domain {1, … , n} induces a natural map from words of length n to those of length m: the ith letter of the output word (1 ≤ i ≤ m) is given by the f(i)th letter of the input word. We study this map in the case where f is a surjection satisfying the condition |f(i+1)-f(i)| ≤ 1 for 1 ≤ i < m. Intuitively, we think of f as describing a "walk" on a word u, visiting every position, and yielding a word w as the sequence of letters encountered en route. If such an f exists, we say that u generates w. Call a word primitive if it is not generated by any word shorter than itself. We show that every word has, up to reversal, a unique primitive generator. Observing that, if a word contains a non-trivial palindrome, it can generate the same word via essentially different walks, we obtain conditions under which, for a chosen pair of walks f and g, those walks yield the same word when applied to a given primitive word. Although the original impulse for studying primitive generators comes from their application to decision procedures in logic, we end, by way of further motivation, with an analysis of the primitive generators for certain word sequences defined via morphisms.