每个字符串的概率自动机复杂度至多为3
Every string has probabilistic automatic complexity at most three
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中文总结 AI 辅助
研究证明任意有限字母表上的所有字符串的概率自动机复杂度至多为3，构造了明确的三状态概率有限自动机，结合已有分类完全确定其在二进制字符串上的取值。
中文摘要 AI 辅助
吉尔（arXiv:2402.13376）引入了有限字符串w的概率自动机复杂度A_P(w)：即概率有限自动机（PFA）的最少状态数，要求w是该长度下被最大概率接受的唯一字符串。他提出A_P是否无界的问题，并指出目前未发现A_P>3的字符串（该论文的问题4.14）。我们通过证明任意有限字母表上的所有字符串w都满足A_P(w)≤3，回答了该问题。所构造的三状态自动机是明确的：其约化动力学跟踪对(u,u²)，其中u是输入的逆基b值，其接受函数为向下抛物线，峰值对应目标字符串的值。结合吉尔对A_P=2的二进制字符串的分类，这完全确定了A_P在二进制字符串上的取值。
英文摘要
Gill (arXiv:2402.13376) introduced the probabilistic automatic complexity $A_P(w)$ of a finite string $w$: the least number of states of a probabilistic finite automaton (PFA) for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, noting that no string with $A_P > 3$ was known (Question 4.14 of that paper). We answer the question by proving that $A_P(w)\le 3$ for every string $w$ over every finite alphabet. The witnessing three-state automaton is explicit: its reduced dynamics tracks the pair $(u,u^2)$, where $u$ is the reversed base-$b$ value of the input, and its acceptance functional is a downward parabola peaked at the value of the target string. Combined with Gill's classification of the binary strings with $A_P=2$, this completely determines $A_P$ on binary strings.
