Given a regular language L over an ordered alphabet $\Sigma$, the set of lexicographically smallest (resp., largest) words of each length is itself regular. Moreover, there exists an unambiguous finite-state transducer that, on a given word w, outputs the length-lexicographically smallest word larger than w (henceforth called the L-successor of w). In both cases, naive constructions result in an exponential blowup in the number of states. We prove that if L is recognized by a DFA with n states, then $2^{\Theta(\sqrt{n \log n})}$ states are sufficient for a DFA to recognize the subset S(L) of L composed of its lexicographically smallest words. We give a matching lower bound that holds even if S(L) is represented as an NFA. We then show that the same upper and lower bounds hold for an unambiguous finite-state transducer that computes L-successors.

中文翻译：

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字典序最小词和计算后继词的状态复杂度

给定有序字母表 $\Sigma$ 上的正则语言 L，每个长度的字典序最小（或最大）单词集本身是正则的。此外，存在一个明确的有限状态转换器，它在给定的单词 w 上输出字典序长度上最小的大于 w 的单词（以下称为 w 的 L-successor）。在这两种情况下，朴素的构造都会导致状态数量呈指数级增长。我们证明，如果 L 被具有 n 个状态的 DFA 识别，那么 $2^{\Theta(\sqrt{n \log n})}$ 状态足以让 DFA 识别 L 的子集 S(L) 由其字典序最小的词。我们给出了一个匹配的下界，即使 S(L) 表示为 NFA。然后，我们证明了相同的上限和下限适用于计算 L 后继的明确有限状态传感器。