We study the hardness of deciding probabilistic termination as well as the hardness of approximating expected values (e.g. of program variables) and (co)variances for probabilistic programs.Termination We distinguish two notions of probabilistic termination: Given a program P and an input $$\sigma $$σ...1....does P terminate with probability 1 on input $$\sigma $$σ? (almost-sure termination)2....is the expected time until P terminates on input $$\sigma $$σ finite? (positive almost-sure termination) For both of these notions, we also consider their universal variant, i.e. given a program P, does P terminate on all inputs? We show that deciding almost-sure termination as well as deciding its universal variant is $$\varPi ^0_2$$Π20-complete in the arithmetical hierarchy. Deciding positive almost-sure termination is shown to be $$\varSigma _2^0$$Σ20-complete, whereas its universal variant is $$\varPi _3^0$$Π30-complete.Expected values Given a probabilistic program P and a random variable f mapping program states to rationals, we show that computing lower and upper bounds on the expected value of f after executing P is $$\varSigma _1^0$$Σ10- and $$\varSigma _2^0$$Σ20-complete, respectively. Deciding whether the expected value equals a given rational value is shown to be $$\varPi ^0_2$$Π20-complete.Covariances We show that computing upper and lower bounds on the covariance of two random variables is both $$\varSigma _2^0$$Σ20-complete. Deciding whether the covariance equals a given rational value is shown to be in $$\varDelta _3^0$$Δ30. In addition, this problem is shown to be $$\varSigma ^0_2$$Σ20-hard as well as $$\varPi ^0_2$$Π20-hard and thus a “proper” $$\varDelta _3^0$$Δ30-problem. All hardness results on covariances apply to variances as well.

中文翻译：

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论分析概率程序的难度

我们研究了决定概率终止的难度以及近似期望值（例如程序变量）和概率程序的（协）方差的难度。终止 我们区分概率终止的两个概念：给定程序 P 和输入 $$ \sigma $$σ...1.... P 在输入 $$\sigma $$σ 上是否以概率 1 终止？（几乎肯定会终止）2.... 直到 P 在输入 $$\sigma $$σ 上终止的预期时间是否有限？（肯定的几乎肯定终止）对于这两个概念，我们还考虑了它们的通用变体，即给定程序 P，P 是否在所有输入上终止？我们表明，在算术层次结构中，决定几乎肯定终止以及决定其通用变体是 $$\varPi ^0_2$$Π20-complete。确定肯定的几乎肯定终止被证明是 $$\varSigma _2^0$$Σ20-complete，而它的通用变体是 $$\varPi _3^0$$Π30-complete。 预期值给定概率程序 P 和随机变量 f 将程序状态映射到有理数，我们表明在执行 P 后计算 f 的期望值的上下界是 $$\varSigma _1^0$$Σ10- 和 $$\varSigma _2^0$$Σ20-分别完成。确定期望值是否等于给定的有理值显示为 $$\varPi ^0_2$$Π20-complete.Covariances 我们表明计算两个随机变量的协方差的上限和下限都是 $$\varSigma _2^ 0$$Σ20-完成。确定协方差是否等于给定的有理值显示在 $$\varDelta _3^0$$Δ30 中。此外，这个问题被证明是 $$\varSigma ^0_2$$Σ20-hard 以及 $$\varPi ^0_2$$Π20-hard，因此是一个“正确的”$$\varDelta _3^0$$Δ30-问题。协方差的所有硬度结果也适用于方差。