最好的时代，最坏的时代：基于矩的概率成本结构分析
The Best of Times, the Worst of Times: Moment-Based Analysis of Probabilistic Cost Structures
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中文总结 AI 辅助
研究含max和min运算的概率程序成本矩的计算问题，提出组合成本分析方法，自下而上通过分层结构求解局部递归方程，用替代分布总结子问题，能计算均值并提升到高阶矩，在工具中实现并经三个问题评估。
中文摘要 AI 辅助
本文研究如何计算某些概率程序成本（如运行时间）的矩（均值、方差等），其中局部成本不仅通过加法组合，还通过极值运算max和min组合。此类成本自然出现，但超出了为加法成本开发的基于矩分析的范围。难点在于max和min是非线性的，仅传播矩会失败，传播全部分布虽足够但计算难处理。我们为一类成本结构可表示为分层成本表达式的概率程序提出了组合成本分析方法。该分析自下而上通过分层结构进行，在每个节点求解局部递归方程并用替代分布总结每个子问题。每个替代分布由精确的短时前缀和紧凑的参数化尾部组成。我们的方法能以合理误差界计算成本分布的均值，并系统地提升到二阶及更高阶矩。此外，可通过细化替代表示提高精度，用额外计算换取更紧的界。我们在工具DICKENS中实现了该方法，并在三个问题上评估了其能力：量子中继器等待时间、RFID冲突解决和fork-join计算的完成时间。
英文摘要
This paper studies how to compute the moments -- mean, variance, and beyond -- of the cost (e.g., running time) of certain probabilistic programs, in which local costs combine not only additively but also via the extremal operations $\max$ and $\min$. Such costs arise naturally -- for instance, the number of rounds of a contention-resolution protocol, the waiting time of a quantum repeater, and the completion time of a fork-join computation -- but fall outside the scope of moment-based analyses developed for additive costs. The difficulty is that $\max$ and $\min$ are nonlinear: the moments of $\max(X, Y)$ are not determined by those of $X$ and $Y$, so propagating moments alone fails. In contrast, propagating full distributions would suffice, but is computationally intractable. We present a compositional cost analysis for a family of probabilistic programs whose cost structure can be represented as a hierarchical cost expression. The analysis proceeds bottom-up through the hierarchical structure, solving local recurrence equations at each node and summarizing each subproblem with a surrogate distribution. Each surrogate consists of an exact short-time prefix and a compact parametric tail. Our approach computes the mean of the cost distribution with a sound error bound, and systematically lifts to second and higher moments. In addition, precision can be increased by refining the surrogate representation, trading additional computation for tighter bounds. We implemented our method in a tool, called DICKENS, and evaluated its capabilities on three problems: quantum repeater waiting times, RFID collision resolution, and completion times of fork-join computations.
