A nonlinear lower bound for constant depth arithmetical circuits via the discrete uncertainty principle

Jansen, Maurice; Regan, Kenneth W.

doi:10.1016/j.tcs.2008.09.029

Cited by 2 publications

(2 citation statements)

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“…There have been very few applications of the discrete uncertainty principle in Computer Science, and in fact we are only familiar with one other such result, concerning circuit lower bounds [13]. We expect that more applications can be found, in particular in cryptography.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Gur¹,

Tamuz²

2013

Chicago J. of Theoretical Comp. Sci.

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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 Ω(k) queries lower bound for this problem.Our proof crucially uses Hirschman's entropic version of Heisenberg's uncertainty principle.

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Section: Introductionmentioning

confidence: 99%

Untitled

Gur¹,

Tamuz²

2013

Chicago J. of Theoretical Comp. Sci.

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“…Our proofs rely on the discrete version of Heisenberg's uncertainty principle. There have been very few applications of the discrete uncertainty principle in Computer Science, and in fact we are only familiar with one other such result, concerning circuit lower bounds [13]. We expect that more applications can be found, in particular in cryptography.…”

Section: Introductionmentioning

confidence: 99%