AI 中文总结
该论文扩展了单词分离相关定理，解决了两个开放问题，证明了单词阶数对分离的无界影响，改进了另一问题的下界。
AI 中文摘要
我们重新探讨了Demaine、Eisenstat、Shallit和Wilson提出的单词分离问题，以及Ebrahimnejad针对他们的逆问题所做的后续研究。对于长度为n的对，其差异单词有d个游程，我们证明了O(d log n)的界，扩展了Demaine等人的汉明距离定理。对于共轭单词，我们给出了由移位算术控制的界。我们解决了Demaine等人的开放问题2，表明两个单词的阶数可使非确定性分离产生无界因子的变化。我们的逆序构造解决了Ebrahimnejad针对开放问题1的后续问题：正向和反向确定性分离可产生无界因子的差异。由于非确定性分离在逆序下不变，同一构造还改进了开放问题3的下界。
英文摘要
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.
Comments7 pages, 2 figures
