A Unified Framework for Testing Linear-Invariant Properties

Abstract

In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed-Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear-invariant properties, drawing on techniques from additive combinatorics and from graph theory.Our main contributions here are the following:1. We introduce a simple combinatorial condition, which we call subspace-heredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area.2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with one-sided error, thus addressing a challenge posed by Sudan recently.3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture.Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.

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“…Bhattacharyya, Grigorescu and Shapira [15] have shown that non-monotone matroid-freeness properties are testable when the matroid is restricted to be of complexity 1. Interestingly, their proof uses arithmetic regularity lemmas, like Green [21] and us, instead of reductions to hypergraph properties as in [28,27,32].…”

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“…However, a general result analogous to Theorem 4.5 remains elusive. For that, as Bhattacharyya, Grigorescu and Shapira [BGS10] show, the following question needs to be resolved.…”

“…Finally, about one-sided testable properties, the analog of Theorem 4.3 is also known. If a property P is obliviously one-sided testable, then it is "semi-subspace-hereditary" [BGS10]. For details, consult the paper, but roughly speaking, a semi-subspace-hereditary property P is close to a subspace-hereditary property P , in the sense that P contains P and any function f : F n → [R] that's ε-far from P is not in P (for large enough n in terms of ε).…”

“…This was borne out by [BGS10] where every affine-invariant property of complexity 1 was showed to be one-sided testable, again through Fourier analysis. The barrier of complexity 1 was crossed in [BFL13] by using higher-order Fourier analysis, although this work restricted itself to properties over finite fields of large characteristic.…”

“…In[BGS10], a positive answer to this question was conjectured. We currently feel unsure about the truth of this, and so we pose the issue as a question.…”