Property Testing via Set-Theoretic Operations

TL;DR

This paper investigates how basic set operations affect the testability of properties, providing new proofs and improved query complexities for specific property testing problems.

Contribution

It systematically studies the testability of property unions, intersections, and differences, and improves query complexity bounds for testing disjunctions of linear functions.

Findings

Linearity testing can be achieved with worse query complexity using new proof techniques.

Disjunction of linear functions can be tested with $O(1/\eps^2)$ queries, improving previous bounds.

Set-theoretic operations on testable properties can preserve testability under certain conditions.

Abstract

Given two testable properties $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, under what conditions are the union, intersection or set-difference of these two properties also testable? We initiate a systematic study of these basic set-theoretic operations in the context of property testing. As an application, we give a conceptually different proof that linearity is testable, albeit with much worse query complexity. Furthermore, for the problem of testing disjunction of linear functions, which was previously known to be one-sided testable with a super-polynomial query complexity, we give an improved analysis and show it has query complexity $O(1/\eps^2)$, where $\eps$ is the distance parameter.