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On the Length of Shortest Strings Accepted by Two-way Finite Automata
Fundamenta Informaticae ( IF 1.166 ) Pub Date : 2021-06-30 , DOI: 10.3233/fi-2021-2044
Egor Dobronravov 1 , Nikita Dobronravov 1 , Alexander Okhotin 1
Affiliation  

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).

中文翻译:

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)。
更新日期:2021-07-04
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