半对抗场景下用自动机区分单词
Separating Words with Automata in the Half-adversarial Case
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中文总结 AI 辅助
本文研究半对抗场景下的DFA单词区分问题，证明对均匀随机二进制单词u，高概率下可用状态数为$O(\n^{7/3} n \log n)$的DFA区分u与任意其他单词，核心方法是利用随机单词结构稀疏性，通过分块压缩将问题归约为短游程编码单词的区分。
中文摘要 AI 辅助
我们研究确定性有限自动机（DFA）的单词区分问题（Goralčík与Koubek，1986）。该问题为：给定两个长度不超过n的不同单词u、v，接受其中一个、拒绝另一个的最小DFA的规模是多少？针对所有长度≤n的单词对，目前最坏情况下的最优上界为$\tilde{O}(n^{1/3})$个状态（Chase，2021），最优下界为$Ω(\n)$。\n 本文研究半随机、半对抗的场景：我们证明，若u是长度为n的均匀随机二进制单词，则在高概率下，对任意不等于u的单词v，存在一个状态数为$O(\n^{7/3} n \log n)$的DFA可区分u和v。我们的结果基于一种利用随机单词结构稀疏性的新颖分析：我们展示了如何通过小型确定性转换器应用分块压缩，将区分问题归约为游程编码较短的单词的情况。
英文摘要
We consider the problem of separating words with deterministic finite automata (DFA) (Goral{č}{í}k and Koubek, 1986). This problem asks: given two distinct words $u,v$ of length at most $n$, what is the size of the smallest DFA that accepts one and rejects the other? The best upper bound on the worst-case over all pairs of words of length at most $n$ is $\tilde{O}(n^{1/3})$ states (Chase, 2021), while the best lower bound is $Ω(\log n)$. In this work, we consider the half-random, half-adversarial case: we show that if $u$ is a uniformly random binary word of length $n$, then with high probability, for any word $v$ not equal to $u$, there is a DFA with $O(\log^{7/3} n \mathrm{poly}\log\log n)$ states that separates $u$ and $v$. Our results are based on a novel analysis that exploits the structural sparsity of random words: we show how to apply block-wise compaction with small deterministic transducers to reduce the separation problem to the case of words with short run-length encodings.
