A characterization of constant-sample testable properties

We characterize the set of properties of Boolean-valued functions f:X ->{0,1} on a finite domain X that are testable with a constant number of samples (x,f(x)) with x drawn uniformly at random from X. Specifically, we show that a property 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 X1, horizontal ellipsis ,Xk of X such that whether f:X ->{0,1} satisfies the property is determined solely by the densities of f on X1, horizontal ellipsis ,Xk. We use this characterization to show that symmetric properties are essentially the only graph properties and affine-invariant properties that are testable with a constant number of samples and that for every constant d >= 1, monotonicity of functions on the d-dimensional hypergrid is testable with a constant number of samples.