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.

翻译：每个非确定型有限自动机识别的语言都可被确定型自动机识别，但代价是状态数可能增加——最坏情况下从$n$个状态增至$2^n$个状态。本文在概率框架下研究这一经典结果：我们随机均匀选取一个含$n$个状态的确定型自动机，并仅添加一条随机转移。这些自动机近乎确定型，因为仅有一个状态在读入字母时存在非确定选择。在我们的模型中，每个状态具有固定的终态概率。我们证明：对任意$d\geq 1$，此类自动机所识别语言的最小（确定型）自动机的状态数以不可忽视的概率超过$n^d$；作为推论，其最小自动机的期望规模增长速度快于任何多项式。当每个状态为终态的概率依赖于$n$但不过分接近$0$或$1$（精确而言距边界至少为$\Omega(\frac1{\sqrt{n}})$）时，该结果依然成立，因而允许期望终态数量为次线性的模型。