概率程序中断言违规的定量分析

Jinyi Wang, Yican Sun, Hongfei Fu, A. K. Goharshady, K. Chatterjee
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引用次数: 13

摘要

我们考虑了一个基本问题:在一个概率规划中,给定的断言被违反的概率的定量边界的推导。我们提供了自动算法来获得断言违反概率的下界和上界。我们的方法的主要新颖之处在于我们证明了新的和专用的不动点定理,这些定理作为我们算法的理论基础,使我们能够根据不动点前和不动点后的函数来推理断言违反边界。为了综合这些不动点,我们设计了利用各种数学工具的算法,包括排斥排序上鞅、Hoeffding引理、Minkowski分解、Jensen不等式和凸优化。在理论方面,我们提供了(i)第一个关于断言违反概率下界的自动算法,(ii)第一个关于仿射程序中指数形式上界的完整算法,以及(iii)可证明且明显比以前的方法更严格的上界。在实践方面,我们展示了我们的算法可以处理文献中各种各样的程序,并且合成的边界比以前的结果要严格得多,在某些情况下可以达到数千个数量级。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Quantitative analysis of assertion violations in probabilistic programs
We consider the fundamental problem of deriving quantitative bounds on the probability that a given assertion is violated in a probabilistic program. We provide automated algorithms that obtain both lower and upper bounds on the assertion violation probability. The main novelty of our approach is that we prove new and dedicated fixed-point theorems which serve as the theoretical basis of our algorithms and enable us to reason about assertion violation bounds in terms of pre and post fixed-point functions. To synthesize such fixed-points, we devise algorithms that utilize a wide range of mathematical tools, including repulsing ranking supermartingales, Hoeffding's lemma, Minkowski decompositions, Jensen's inequality, and convex optimization. On the theoretical side, we provide (i) the first automated algorithm for lower-bounds on assertion violation probabilities, (ii) the first complete algorithm for upper-bounds of exponential form in affine programs, and (iii) provably and significantly tighter upper-bounds than the previous approaches. On the practical side, we show our algorithms can handle a wide variety of programs from the literature and synthesize bounds that are remarkably tighter than previous results, in some cases by thousands of orders of magnitude.
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