Property Testing via Set-Theoretic Operations

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.

💡 Research Summary

The paper investigates a fundamental question in property testing: given two testable properties 𝒫₁ and 𝒫₂, under what circumstances do their set‑theoretic combinations—union, intersection, and set‑difference—remain testable? The authors develop a systematic framework that establishes sufficient conditions for each operation, and they demonstrate how these conditions can be leveraged to design new testers for composite properties.

For unions, the authors show that if each property admits an (ε,δ)-tester, then running the two testers independently on the same input and accepting if either accepts yields a tester for 𝒫₁∪𝒫₂ with only a constant factor increase in query complexity. The key technical step is error amplification: by repeating the individual testers a bounded number of times, the overall error probability can be kept below the desired ε.

Intersections are more delicate because both conditions must hold simultaneously. The paper introduces a “joint sampling” technique: the same random sample set is fed to both testers, and the acceptance decision is the logical AND of their outcomes. Under the assumption that the two properties are independent (or that one provides a strong one‑sided guarantee), this approach preserves testability with query complexity essentially the sum of the individual complexities.

Set‑differences 𝒫₁\𝒫₂ require a negative tester for 𝒫₂. The authors prove that if 𝒫₂ has a one‑sided tester that never rejects a member of 𝒫₂ but rejects non‑members with high probability, then 𝒫₁\𝒫₂ is testable by combining the tester for 𝒫₁ with the complement of the one‑sided tester for 𝒫₂. The resulting query complexity is the additive combination of the two component testers.

To illustrate the utility of their framework, the authors present two applications. First, they give a novel proof that linearity is testable. By decomposing linearity into two simpler, testable sub‑properties (average‑preservation and a bilinear condition) and applying the intersection result, they obtain a tester for linearity. This construction is conceptually clean but incurs a query complexity of O(1/ε²), which is worse than the classic BLR tester’s O(1/ε). The gap highlights that while set‑theoretic composition guarantees existence, it does not automatically yield optimal efficiency.

Second, the paper revisits the problem of testing the disjunction of linear functions. Prior work established one‑sided testability but with super‑polynomial query complexity. Using the set‑difference condition together with a refined probabilistic analysis, the authors design a tester that samples random linear combinations, checks each candidate linear function independently, and aggregates the results. This yields a dramatically improved query complexity of O(1/ε²).

Overall, the work provides a clear set‑theoretic lens for understanding when complex properties inherit testability from their components. It establishes general sufficient conditions for union, intersection, and difference, and shows how these can be applied to both classic and newly studied testing problems. The results open avenues for further research on necessary conditions, tighter bounds, and the design of more query‑efficient testers for composite properties.