Abstract
For a class \({\mathcal{F}}\) of formulas (general de Morgan or read-once de Morgan), the shrinkage exponent \({\Gamma_{\mathcal{F}}}\) is the parameter measuring the reduction in size of a formula \({F\in\mathcal{F}}\) after \({F}\) is hit with a random restriction. A Boolean function \({f\colon \{0,1\}^n\to\{1,-1\}}\) is Fourier-concentrated if, when viewed in the Fourier basis, \({f}\) has most of its total mass on “low-degree” coefficients. We show a direct connection between the two notions by proving that shrinkage implies Fourier concentration: For a shrinkage exponent \({\Gamma_{\mathcal{F}}}\), a formula \({F\in\mathcal{F}}\) of size \({s}\) will have most of its Fourier mass on the coefficients of degree up to about \({s^{1/\Gamma_{\mathcal{F}}}}\). More precisely, for a Boolean function \({f\colon\{0,1\}^n\to\{1,-1\}}\) computable by a formula of (large enough) size \({s}\) and for any parameter \({r > 0}\),
where \({\Gamma}\) is the shrinkage exponent for the corresponding class of formulas: \({\Gamma=2}\) for de Morgan formulas, and \({\Gamma=1/\log_2(\sqrt{5}-1)\approx 3.27}\) for read-once de Morgan formulas. This Fourier concentration result is optimal, to within the \({o(1)}\) term in the exponent of \({s}\).
As a standard application of these Fourier concentration results, we get that subquadratic-size de Morgan formulas have negligible correlation with parity. We also show the tight \({\Theta(s^{1/\Gamma})}\) bound on the average sensitivity of read-once formulas of size \({s}\), which mirrors the known tight bound \({\Theta(\sqrt{s})}\) on the average sensitivity of general de Morgan \({s}\)-size formulas.
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Impagliazzo, R., Kabanets, V. Fourier Concentration from Shrinkage. comput. complex. 26, 275–321 (2017). https://doi.org/10.1007/s00037-016-0134-y
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DOI: https://doi.org/10.1007/s00037-016-0134-y
Keywords
- formula complexity
- random restrictions
- de Morgan formulas
- read-once de Morgan formulas
- shrinkage exponent
- Fourier analysis of Boolean functions
- Fourier concentration
- average sensitivity