AI 中文总结
研究针对k-不同语言突破$4^k$障碍的问题，核心方法是将经典要素组织成小工具放大框架并引入组合与压缩技术，主要贡献是给出大小为$3.918^k n^{O(1)}$的非确定性有限自动机。
AI 中文摘要
对于整数$k\leq n$，令$L_{k,n}$为长度至多为k且无符号重复的[n]上的单词集合。我们给出了一个大小为$3.918^k n^{O(1)}$的非确定性有限自动机（NFA），改进了Ben-Basat、Gabizon和Zehavi的$4^{k + o(k)}n^{O(1)}$构造。我们的证明将乘积自动机、哈希和系数估计等经典要素组织成一个小工具放大框架。通过对一个小的局部NFA小工具的多个副本进行乘积，并将k个输入符号哈希到副本和局部颜色。哈希族保证对于每个无重复输入，某些哈希将至多r个符号发送到每个副本，使得每个副本中的投影被局部小工具接受。取相应乘积NFA的非确定性并集得到全局NFA。通过放大从小型维特设计$S(4,5,11)$获得的用于$L_{6,11}$的200状态小工具，该框架给出了一个大小为$3.967^k n^{O(1)}$的NFA。然后我们引入组合与压缩技术，删除这些乘积中昂贵的中间层，并用合理的单符号快捷转换替换跨越删除带的路径。我们应用它两次，一次用于增强放大框架，一次用于局部小工具，从而得到所述结果。
英文摘要
For integers $k\le n$, let $L_{k,n}$ be the set of words over $[n]$ of length at most $k$ in which no symbol is repeated. We present a nondeterministic finite automaton (NFA) of size $3.918^k n^{O(1)}$, improving on the $4^{k+o(k)}n^{O(1)}$ construction of Ben-Basat, Gabizon, and Zehavi. Our proof organizes several classical ingredients---product automata, hashing, and coefficient estimates---into a gadget-amplification framework: We take the product of many copies of a small local NFA gadget, whose language is a subset of $L_{r,c}$, and hash the $k$ input symbols to copies and local colors. The hash family guarantees that, for every repetition-free input, some hash sends at most $r$ symbols to each copy such that the resulting projection in every copy is accepted by the local gadget. Taking the nondeterministic union of the corresponding product NFAs yields a global NFA. Amplifying a $200$-state gadget for $L_{6,11}$ obtained from the small Witt design $S(4,5,11)$, this framework gives a $3.967^k n^{O(1)}$-size NFA. We then introduce the compose-and-compress technique, which deletes the expensive middle layers of these products and replaces paths across the deleted bands with sound one-symbol shortcut transitions. We apply it twice, once for enhancing the amplification framework and again for the local gadget, obtaining the stated result.
