A Characterization of Constant-Sample Testable Properties

by Eric Blais and Yuichi Yoshida

Oded's comments

Characterizations of problems according to their complexity are a cherished goal of TOC, even when one considers very lowe levels of complexity as is the case with sample-based testability with a constant number of samples. This work articulates the intuition that this complexity class is indeed very restricted, and indicates exacly what it can do.

The original abstract

We characterize the set of properties of Boolean-valued functions on a finite domain $\mathcal{X}$ that are testable with a constant number of samples. Specifically, we show that a property $\mathcal{P}$ is testable with a constant number of samples if and only if it is (essentially) a $k$-part symmetric property for some constant $k$, where a property is $k$-part symmetric if there is a partition $S_1,\ldots,S_k$ of $\mathcal{X}$ such that whether $f:\mathcal{X} \to \{0,1\}$ satisfies the property is determined solely by the densities of $f$ on $S_1,\ldots,S_k$.

We use this characterization to obtain a number of corollaries, namely: (i) A graph property $\mathcal{P}$ is testable with a constant number of samples if and only if whether a graph $G$ satisfies $\mathcal{P}$ is (essentially) determined by the edge density of $G$. (ii) An affine-invariant property $\mathcal{P}$ of functions $f:\mathbb{F}_p^n \to \{0,1\}$ is testable with a constant number of samples if and only if whether $f$ satisfies $\mathcal{P}$ is (essentially) determined by the density of $f$. (iii) For every constant $d \geq 1$, monotonicity of functions $f : [n]^d \to \{0, 1\}$ on the $d$-dimensional hypergrid is testable with a constant number of samples.

See ECCC TR16-201.


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