Abstract
AbstractWe study the probabilistic degree over of the OR function on n variables. For , the ‐error probabilistic degree of any Boolean function f : {0, 1}n → {0, 1} over is the smallest nonnegative integer d such that the following holds: there exists a distribution P of polynomials of degree at most d such that for all , we have . It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the ‐error probabilistic degree of the OR function is at most . Our first observation is that this can be improved to which is better for small values of . In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution P have the following special structure: where each Li(x1, … , xn) is a linear form in the variables x1, … , xn, that is, the polynomial is a product of affine forms. We show that the ‐error probabilistic degree of OR when restricted to polynomials of the above form is , thus matching the above upper bound (up to poly‐logarithmic factors).
Highlights
We show that the ε-error probabilistic degree of OR when restricted to polynomials of the above form is Ω
Low-degree polynomial approximations of Boolean functions were introduced by Razborov in his celebrated work [11] on proving lower bounds for the class of Boolean functions computed by low-depth circuits
The ε-error Probabilistic degree of f, denoted by P-degε(f ), is the smallest non-negative integer d such that the following holds: there exists an ε-error probabilistic polynomial P over R such that P is entirely supported on polynomials of degree at most d
Summary
Low-degree polynomial approximations of Boolean functions were introduced by Razborov in his celebrated work [11] on proving lower bounds for the class of Boolean functions computed by low-depth circuits. Tarui [14] and Beigel et al [3] and the exact representation of degree n mentioned above Given this observation, one might hope to prove a matching lower bound on the ε-error probabilistic degree of ORn. We can show such a bound (upto polylogarithmic factors) if we suitably restrict the class of polynomials being considered. For this class of polynomials, we prove the following (almost) tight result on the ε-error hyperplane covering probabilistic degree of the OR function. For any any positive integer n and ε ∈ (0, 1/3), hcP-degε(ORn) It is open if this result can be extended to prove a tighter lower bound on the ε-error probabilistic degree of the ORn function. We remark that though our lower bound works for a larger class of polynomials, our proof technique is inspired by their proof
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