A Theory AB Toolbox

Marco Gaboardi, Justin Hsu
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Abstract

Randomized algorithms are a staple of the theoretical computer science literature. By careful use of randomness, algorithms can achieve properties that are simply not possible with deterministic algorithms. Today, these properties are proved on paper, by theoretical computer scientists; we investigate formally verifying these proofs. The main challenges are two: proofs about algorithms can be quite complex, using various facts from probability theory; and proofs are highly customized - two proofs of the same property for two algorithms can be completely different. To overcome these challenges, we propose taking inspiration from paper proofs, by building common tools - abstractions, reasoning principles, perhaps even notations - into a formal verification toolbox. To give an idea of our approach, we consider three common patterns in paper proofs: the union bound, concentration bounds, and martingale arguments.
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AB理论工具箱
随机算法是理论计算机科学文献的主要内容。通过谨慎地使用随机性,算法可以实现确定性算法根本无法实现的特性。今天,理论计算机科学家在纸上证明了这些特性;我们研究正式验证这些证明。主要的挑战有两个:关于算法的证明可能非常复杂,需要使用概率论中的各种事实;而且证明是高度自定义的——两个算法的相同性质的两个证明可以完全不同。为了克服这些挑战,我们建议从纸质证明中获得灵感,通过构建通用工具——抽象、推理原则,甚至可能是符号——到正式的验证工具箱中。为了说明我们的方法,我们考虑了纸上证明的三种常见模式:联合界、集中界和鞅论证。
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