置换自动机分离单词的一个改进下界
An Improved Lower Bound for Separating Words by Permutation Automata
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中文总结 AI 辅助
该研究改进了置换自动机分离二元单词的状态数下界，通过构造对称群恒等式的正单词对，将原下界提升至Ω(log n log log n)，并补充了更精确的常数与低阶项。
中文摘要 AI 辅助
我们证明，对于每个足够大的规定长度n的二元单词对，需要Ω(log n log log n)个状态才能由字母作为置换的确定有限自动机分离。我们通过构造长度为exp(O(k/log k))的二元正单词对，使其构成对称群𝔖_k的恒等式，从而得到该下界。这改进了Bulatov、Karpova、Shur和Startsev（arXiv:1609.03199）之前的下界(3/2 - o(1))log n。该证明结合了正单词构造与置换阶覆盖，素数幂的连续段将覆盖问题简化为枚举和有界的正整数集合。所需估计使用中心二项式系数和不同部分分拆的经典渐近公式，附录通过贪心覆盖和定量素数定理给出更精确的常数和低阶项。
英文摘要
We prove that some pairs of binary words of every sufficiently large prescribed length $n$ require $Ω(\log n\log\log n)$ states to separate by a deterministic finite automaton whose letters act as permutations. We obtain this bound by constructing distinct equal-length positive binary words that form an identity of $\mathfrak{S}_k$ of length $\exp(O(k/\log k))$. This improves the previous lower bound $(3/2-o(1))\log n$ of Bulatov, Karpova, Shur, and Startsev (arXiv:1609.03199). The proof combines a positive-word construction with a cover of permutation orders. Consecutive segments of prime powers reduce the covering problem to enumerating sets of positive integers with bounded sum. The required estimates use the central binomial coefficient and the classical asymptotic formula for partitions into distinct parts. An appendix gives sharper constants and lower-order terms using a greedy cover and a quantitative prime number theorem.
