Given a two-way finite automaton recognizing a non-empty language, consider the length of the shortest string it accepts, and, for each n ≥ 1, let f(n) be the maximum of these lengths over all n-state automata. It is proved that for n-state two-way finite automata, whether deterministic or nondeterministic, this number is at least Ω(10n/5) and less than (2nn+1), with the lower bound reached over an alphabet of size Θ(n). Furthermore, for deterministic automata and for a fixed alphabet of size m ≥ 1, the length of the shortest string is at least e(1+o(1))mn(log n− log m).

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关于双向有限自动机接受的最短字符串的长度

给定一个识别非空语言的双向有限自动机，考虑它接受的最短字符串的长度，并且，对于每个 n ≥ 1，让 f(n) 是所有 n 状态自动机的这些长度中的最大值。证明对于 n 状态双向有限自动机，无论是确定性还是非确定性，该数字至少为 Ω(10n/5) 且小于 (2nn+1)，下界达到了大小为 Θ 的字母表(n)。此外，对于确定性自动机和大小为 m ≥ 1 的固定字母表，最短字符串的长度至少为 e(1+o(1))mn(log n− log m)。