AI 中文总结
本文研究一元语言的二元编码版本在双向自动机（2DFA）下的状态数变化，给出转换上界、特殊情况上界及下界，并构造n≥7时的一元见证语言验证相关结论。
AI 中文摘要
若L是一元语言，则其二元编码版本bin(L)是包含所有代表L中任意0^x的二进制串的二元语言。已知若一元语言L是正则的，且可被状态数为n的最小单向确定有限自动机（1DFA）识别，则其二元编码版本也正则，且可被状态数至多为n、至少为1+log(n)的1DFA识别。本文给出双向自动机（2DFA）的相关结果：首先证明每个状态数为n的一元2DFA A'可转换为识别bin(L(A'))的2DFA A''，其状态数至多为O(n)；若A'是最小的且仅使用奇数长度的循环，则A''的状态数至多为2n+2，但至少为n。对于每个n≥7，本文还给出一个一元见证语言，其最小2DFA恰好使用n个状态，但任何识别其二元编码版本的最小2DFA至少使用n个状态，且少于n+log(n)个状态。
英文摘要
If L is a unary language, then its binary coded version bin(L) is a binary language containing all binary strings representing any 0^x in L. It is known that if a unary language L is regular and can be recognized by a minimal one-way deterministic finite automaton (1DFA) with n states, then its binary coded version is also regular and can be recognized by a 1DFA with at most n states, but at least 1+log(n) states. Here we shall present related results for two-way automata (2DFAs). First, we shall show that each unary 2DFA A' with n states can be converted to a 2DFA A'' recognizing bin(L(A')) with at most O(n.log n) states. If A' is minimal and uses only loops of odd lengths, A'' will use at most 2n+2 states, but it must use at least n states. For each n>=7, we shall also present a unary witness language for which a minimal 2DFA uses exactly n states, but any minimal 2DFA recognizing its binary coded version uses at least n states, but less than n+log(n) states.
CommentsIn Proceedings AFL 2026, arXiv:2608.23071
Journal refEPTCS 451, 2026, pp. 1-16
