Testing Linear-Invariant Non-Linear Properties

We c onsider the task of testing properties of Boolean functions that are invari- ant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for "triangle freeness": A function f : F n ! F2 satisfies this property if f(x), f(y), f(x+ y) do not all equal 1, for any pair x, y 2 F n. Here we extend this test to a more systematic study of testing for linear-invariant non- linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by k points v1, . . . , vk 2 F k and f : F n ! F2 satisfies the property that if for all linear maps L : F k ! F n it is the case that f(L(v1)), . . . , f(L(vk)) do not all equal 1. We show that this property is testable if the underlying matroid specified by v1, . . . , vk is a graphic matroid. This extends Green's result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of "1-complexity linear systems" of Green and Tao, and graphic matroids, to derive the results.

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