Further Remarks on Separating Words

Abstract

We revisit questions on separating words raised by Demaine, Eisenstat, Shallit, and Wilson, together with Ebrahimnejad's follow-up to their reversal problem. For length-$n$ pairs whose difference word has $d$ runs, we prove an $O(d\log n)$ bound, extending the Hamming-distance theorem of Demaine et al. For conjugate words, we give bounds controlled by the arithmetic of the shift. We resolve Demaine et al.'s Open Problem 2 by showing that the order of two words can change nondeterministic separation by an unbounded factor. Our reversal construction addresses Ebrahimnejad's follow-up to Open Problem 1: forward and reversed deterministic separation can differ by an unbounded factor. Since nondeterministic separation is invariant under reversal, the same construction also improves the lower bound in Open Problem 3.