# A Semantic Account of Metric Preservation

## TL;DR

This paper develops a denotational semantics framework using metric CPOs to analyze program sensitivity, extending the understanding of metric preservation in probabilistic and differential privacy contexts.

## Contribution

It introduces metric CPOs as a new semantic structure for reasoning about program sensitivity and provides a model for the deterministic fragment of Fuzz.

## Key findings

- Metric CPOs enable reasoning about metric properties of programs.
- The model demonstrates how metric preservation can be understood denotationally.
- Supports analysis of program robustness in privacy and cyber-physical systems.

## Abstract

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 $r \cdot d$. 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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Source: https://tomesphere.com/paper/1702.00374