In this article, we consider the termination problem of probabilistic programs with real-valued variables. The questions concerned are: qualitative ones that ask (i) whether the program terminates with probability 1 (almost-sure termination) and (ii) whether the expected termination time is finite (finite termination); and quantitative ones that ask (i) to approximate the expected termination time (expectation problem) and (ii) to compute a bound B such that the probability not to terminate after B steps decreases exponentially (concentration problem). To solve these questions, we utilize the notion of ranking supermartingales, which is a powerful approach for proving termination of probabilistic programs. In detail, we focus on algorithmic synthesis of linear ranking-supermartingales over affine probabilistic programs (A pps ) with both angelic and demonic non-determinism. An important subclass of A pps is LRA pp which is defined as the class of all A pps over which a linear ranking-supermartingale exists. Our main contributions are as follows. Firstly, we show that the membership problem of LRA pp (i) can be decided in polynomial time for A pps with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for A pps with angelic non-determinism. Moreover, the NP-hardness result holds already for A pps without probability and demonic non-determinism. Secondly, we show that the concentration problem over LRA pp can be solved in the same complexity as for the membership problem of LRA pp . Finally, we show that the expectation problem over LRA pp can be solved in 2EXPTIME and is PSPACE-hard even for A pps without probability and non-determinism (i.e., deterministic programs). Our experimental results demonstrate the effectiveness of our approach to answer the qualitative and quantitative questions over A pps with at most demonic non-determinism.

中文翻译：

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仿射概率程序的定性和定量终止问题的算法分析

在本文中，我们考虑具有实值变量的概率程序的终止问题。所涉及的问题是：定性问题，询问 (i) 程序是否以概率 1 终止（几乎肯定终止）和 (ii) 预期终止时间是否是有限的（有限终止）；和定量的要求（i）近似预期终止时间（预期问题）和（ii）计算界限乙使得在之后不终止的概率乙步数呈指数下降（集中问题）。为了解决这些问题，我们利用了排名超级鞅的概念，这是一种证明概率程序终止的强大方法。具体来说，我们专注于仿射概率程序（Apps) 具有天使和恶魔的非决定论。A的一个重要子类pps是上帝军pp定义为所有 A 的类pps在其上存在线性排名-上鞅。我们的主要贡献如下。首先，我们证明了 LRA 的成员问题pp(i) 可以在多项式时间内确定 Apps至多具有恶魔的不确定性，并且 (ii) 是 NP-hard 并且在 PSPACE 中对于 Apps带着天使般的不确定性。此外，NP 硬度结果已经适用于 Apps没有概率和恶魔的不确定性。其次，我们证明了 LRA 上的集中问题pp可以以与 LRA 的成员资格问题相同的复杂性来解决pp. 最后，我们证明了 LRA 上的期望问题pp可以在 2EXPTIME 中解决，即使对于 A 也是 PSPACE-hardpps没有概率和非确定性（即确定性程序）。我们的实验结果证明了我们的方法在回答 A 上的定性和定量问题的有效性pps最多带有恶魔般的不确定性。