精确非确定性自动机复杂度与唯一非确定性自动机复杂度
Exact versus unique nondeterministic automatic complexity
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中文总结 AI 辅助
该研究证明精确非确定性自动机复杂度$A_{Ne}$与唯一非确定性自动机复杂度$A_N$存在差异，找到二进制分离单词w，经穷举搜索和SAT求解器验证，分离结果由AI智能体发现。
中文摘要 AI 辅助
单词x的精确非确定性自动机复杂度$A_{Ne}(x)$是接受x且不接受其他任何长度为|x|的单词的非确定性有限自动机的最小状态数；唯一非确定性自动机复杂度$A_N(x)$额外要求接受计算是唯一的。Chen、Kjos-Hanssen、Koswara、Richter和Stephan（FSTTCS 2025）提出这两种度量是否存在差异的问题。我们给出肯定回答：二进制单词$w=1101000000100$满足$A_{Ne}(w)=6<7=A_N(w)$。该分离由一个六状态自动机证明，其唯一接受的长度为13的单词是w，且通过恰好三条计算路径接受。匹配的下界通过穷举搜索和SAT求解器确定，不可满足性证明以DRAT格式验证。穷举统计显示，13是任意字母表（二进制或非二进制）上分离单词的最小长度，且经字母重命名后，长度为13的分离单词恰好是单个七字母单词0123444445126的365种字母合并，所有这些合并的$A_{Ne}=6<7=A_N$；其中恰好8种是二进制的。分离单词和验证核心由一个人工智能智能体找到。
英文摘要
The exact nondeterministic automatic complexity $A_{Ne}(x)$ of a word $x$ is the minimum number of states of a nondeterministic finite automaton that accepts $x$ and no other word of length $|x|$; the unique nondeterministic automatic complexity $A_N(x)$ additionally requires the accepting computation to be unique. Chen, Kjos-Hanssen, Koswara, Richter, and Stephan (FSTTCS 2025) asked whether the two measures can differ. We answer this affirmatively: the binary word $w=1101000000100$ satisfies $A_{Ne}(w)=6<7=A_N(w)$. The separation is witnessed by a six-state automaton whose sole accepted word of length $13$ is $w$, accepted along exactly three computation paths. The matching lower bounds were established by exhaustive search and by SAT solvers, with proofs of unsatisfiability certified in the DRAT format. An exhaustive census shows that $13$ is the least length of a separating word over any alphabet, binary or not, and that up to renaming of letters the separating words of length $13$ are precisely $365$ letter-mergings of the single seven-letter word $0123444445126$, all with $A_{Ne}=6<7=A_N$; exactly eight of them are binary. The separating word and the core of the verification were found by an artificial intelligence agent.
