度量保持的语义解释

Arthur Azevedo de Amorim, Marco Gaboardi, Justin Hsu, Shin-ya Katsumata, Ikram Cherigui
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引用次数: 46

摘要

程序敏感性衡量程序对其输入的微小变化的鲁棒性,是从差分隐私到网络物理系统等领域的基本概念。形式化程序灵敏度的一种自然方法是根据输入和输出空间的度量,要求r敏感函数将距离为d的输入映射到距离最多为r·d的输出。因此,程序灵敏度是程序的Lipschitz连续性的模拟。Reed和Pierce介绍了Fuzz,这是一种函数式语言,具有线性类型系统,可以表达程序的灵敏度。它们以度量保存属性的形式在操作上显示出可靠性。受他们工作的启发,我们从外延的角度研究程序敏感性和度量保存。特别地,我们通过赋予度量cpo一个兼容的距离概念,引入了度量cpo——一种新的用于度量空间计算推理的语义结构。这种结构对于程序的度量性质,特别是程序的灵敏度的推理是有用的。我们通过给出模糊的确定性片段的模型来证明度量CPOs。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A semantic account of metric preservation
Program sensitivity measures how robust a program is to small changes in its input, and is a fundamental notion in domains ranging from differential privacy to cyber-physical systems. A natural way to formalize program sensitivity is in terms of metrics on the input and output spaces, requiring that an r-sensitive function map inputs that are at distance d to outputs that are at distance at most r · d. Program sensitivity is thus an analogue of Lipschitz continuity for programs. Reed and Pierce introduced Fuzz, a functional language with a linear type system that can express program sensitivity. They show soundness operationally, in the form of a metric preservation property. Inspired by their work, we study program sensitivity and metric preservation from a denotational point of view. In particular, we introduce metric CPOs, a novel semantic structure for reasoning about computation on metric spaces, by endowing CPOs with a compatible notion of distance. This structure is useful for reasoning about metric properties of programs, and specifically about program sensitivity. We demonstrate metric CPOs by giving a model for the deterministic fragment of Fuzz.
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