Probabilistic automata are an extension of nondeterministic finite automata in which transitions are annotated with probabilities. Despite its simplicity, this model is very expressive and many algorithmic questions are undecidable. In this work we focus on the emptiness problem (and its variant the value problem), which asks whether a given probabilistic automaton accepts some word with probability greater than a given threshold. We consider finitely ambiguous probabilistic automata.

Our main contributions are to construct efficient algorithms for analysing finitely ambiguous probabilistic automata through a reduction to a multi-objective optimisation problem called the stochastic path problem. We obtain a polynomial time algorithm for approximating the value of probabilistic automata of fixed ambiguity and a quasi-polynomial time algorithm for the emptiness problem for 2-ambiguous probabilistic automata.

We complement these positive results by an inapproximability result stating that the value of finitely ambiguous probabilistic automata cannot be approximated unless $\mathbf{P}=\mathbf{NP}$.

概率自动机是非确定性有限自动机的扩展，其中转换用概率注释。尽管它很简单，但这个模型非常具有表现力，许多算法问题是不可判定的。在这项工作中，我们关注空性问题（及其变体值问题），它询问给定的概率自动机是否接受某个概率大于给定阈值的单词。我们考虑有限模糊的概率自动机。

我们的主要贡献是通过归约到称为随机路径问题的多目标优化问题来构建用于分析有限模糊概率自动机的有效算法。我们获得了近似固定模糊概率自动机值的多项式时间算法和用于2歧义概率自动机的空性问题的拟多项式时间算法。

我们通过不可逼近性结果补充这些正面结果，说明有限模糊概率自动机的值不能被逼近，除非 $\mathbf{磷}=\mathbf{NP}$.