一句话：证明在随机 DFA 上加入一条随机转移后，其语言的最小 DFA 期望规模可超过任意多项式。
本文研究非确定性导致确定化膨胀的经典问题：NFA 转为 DFA 最坏可从 n 个状态膨胀到 2^n 个状态。作者考察一种概率模型：均匀随机取一个 n 状态 DFA，再随机加入一条转移，故只有某状态在某字母上存在非确定选择；每状态以固定概率为接受态。通过概率与渐近分析，证明对任意 d≥1，该自动机识别语言的最小 DFA 以不可忽略概率超过 n^d 个状态，因而期望规模比任意多项式增长更快。该结果也适用于接受概率随 n 变化的情形，只要其与 0 和 1 的距离至少为 Ω(1/√n)，从而允许期望接受态数为亚线性。
Every language recognized by a non-deterministic finite automaton can be recognized by a deterministic automaton, at the cost of a potential increase of the number of states, which in the worst case can go from $n$ states to $2^n$ states. In this article, we investigate this classical result in a probabilistic setting where we take a deterministic automaton with $n$ states uniformly at random and add just one random transition. These automata are almost deterministic in the sense that only one state has a non-deterministic choice when reading an input letter. In our model, each state has a fixed probability to be final. We prove that for any $d\geq 1$, with non-negligible probability the minimal (deterministic) automaton of the language recognized by such an automaton has more than $n^d$ states; as a byproduct, the expected size of its minimal automaton grows faster than any polynomial. Our result also holds when each state is final with some probability that depends on $n$, as long as it is not too close to $0$ and $1$, at distance at least $\Omega(\frac1{\sqrt{n}})$ to be precise, therefore allowing models with a sublinear number of final states in expectation.
DOI 原文 · arXiv · PDF（开放获取） · DBLP
@article{CarayolDKN25,
title = {Random Deterministic Automata With One Added Transition},
author = {Arnaud Carayol and Philippe Duchon and Florent Koechlin and Cyril Nicaud},
journal = {Logical Methods in Computer Science},
volume = {Volume 21, Issue 1},
year = {2025},
doi = {10.46298/lmcs-21(1:11)2025}
}