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Let $\mathcal{P}$ be a property of function $\mathbb{F}_p^n \to \{0,1\}$ fora fixed prime $p$. An algorithm is called a tester for $\mathcal{P}$ if, givena query access to the input function $f$, with high probability, it acceptswhen $f$ satisfies $\mathcal{P}$ and rejects when $f$ is "far" from satisfying$\mathcal{P}$. In this paper, we give a characterization of affine-invariantproperties that are (two-sided error) testable with a constant number ofqueries. The characterization is stated in terms of decomposition theorems,which roughly claim that any function can be decomposed into a structured partthat is a function of a constant number of polynomials, and a pseudo-randompart whose Gowers norm is small. We first give an algorithm that tests whetherthe structured part of the input function has a specific form. Then we showthat an affine-invariant property is testable with a constant number of queriesif and only if it can be reduced to the problem of testing whether thestructured part of the input function is close to one of a constant number ofcandidates.

A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems