11,238 research outputs found

    Testing Low Complexity Affine-Invariant Properties

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    Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the Reed-Muller code over F_p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials except that low degree is preserved by composition with affine maps. The complexity of an affine-invariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p >=2 and fixed integer R >= 2, any affine-invariant property P of functions f: F_p^n -> [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graph-theoretic techniques in an algebraic setting, using tools from higher-order Fourier analysis.Comment: 38 pages, appears in SODA '1

    Every locally characterized affine-invariant property is testable

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    Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties

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

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    Let P\mathcal{P} be a property of function Fpn→{0,1}\mathbb{F}_p^n \to \{0,1\} for a fixed prime pp. An algorithm is called a tester for P\mathcal{P} if, given a query access to the input function ff, with high probability, it accepts when ff satisfies P\mathcal{P} and rejects when ff is "far" from satisfying P\mathcal{P}. In this paper, we give a characterization of affine-invariant properties that are (two-sided error) testable with a constant number of queries. The characterization is stated in terms of decomposition theorems, which roughly claim that any function can be decomposed into a structured part that is a function of a constant number of polynomials, and a pseudo-random part whose Gowers norm is small. We first give an algorithm that tests whether the structured part of the input function has a specific form. Then we show that an affine-invariant property is testable with a constant number of queries if and only if it can be reduced to the problem of testing whether the structured part of the input function is close to one of a constant number of candidates.Comment: 27 pages, appearing in STOC 2014. arXiv admin note: text overlap with arXiv:1306.0649, arXiv:1212.3849 by other author

    Testing Linear-Invariant Non-Linear Properties

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    We consider the task of testing properties of Boolean functions that are invariant 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:\cube^{n}\to\cube satisfies this property if f(x),f(y),f(x+y)f(x),f(y),f(x+y) do not all equal 1, for any pair x,y\in\cube^{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 kk points v_{1},...,v_{k}\in\cube^{k} and f:\cube^{n}\to\cube satisfies the property that if for all linear maps L:\cube^{k}\to\cube^{n} it is the case that f(L(v1)),...,f(L(vk))f(L(v_{1})),...,f(L(v_{k})) do not all equal 1. We show that this property is testable if the underlying matroid specified by v1,...,vkv_{1},...,v_{k} 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.Comment: This is the full version; conference version appeared in the proceedings of STACS 200
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