We present the first study of non-deterministic weighted automata under probabilistic semantics. In this semantics words are random events, generated by a Markov chain, and functions computed by weighted automata are random variables. We consider the probabilistic questions of computing the expected value and the cumulative distribution for such random variables.

The exact answers to the probabilistic questions for non-deterministic automata can be irrational and are uncomputable in general. To overcome this limitation, we propose approximation algorithms for the probabilistic questions, which work in exponential time in the size of the automaton and polynomial time in the size of the Markov chain and the given precision. We apply this result to show that non-deterministic automata can be effectively determinised with respect to the standard deviation metric.

我们介绍了概率语义下非确定性加权自动机的首次研究。在这种语义中，单词是由马尔可夫链生成的随机事件，而加权自动机计算的函数是随机变量。我们考虑计算此类随机变量的期望值和累积分布的概率问题。

对于不确定性自动机的概率性问题的确切答案可能是不合理的，并且通常是无争议的。为了克服此限制，我们为概率问题提出了一种近似算法，该算法在自动机大小的指数时间内工作，在马尔可夫链大小和给定精度下的多项式时间工作。我们应用此结果表明，相对于标准差度量，可以有效地确定非确定性自动机。