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arXiv · 1204.0944

Testing Booleanity and the Uncertainty Principle

Tom Gur·Omer Tamuz
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Abstract

Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in {-1,1}. We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean. In particular, we show that a multilinear polynomial with at most k terms must either be Boolean, or output values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its domain. It follows that given oracle access to f, together with the guarantee that its representation as a multilinear polynomial has at most k terms, one can test Booleanity using O(k^2) queries. We show an \Omega(k) queries lower bound for this problem. Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle.

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Tom Gur, Omer Tamuz. 2012-04-04. Testing Booleanity and the Uncertainty Principle. https://arxiv.org/abs/1204.0944

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