Paper Info
Title
On the Probabilistic Degree of OR over the Reals.
Abstract
We study the probabilistic degree over R of the OR function on n variables. For epsilon in (0,1/3), the epsilon-error probabilistic degree of any Boolean function f:{0,1}^n -u003e {0,1} over R is the smallest non-negative integer d such that the following holds: there exists a distribution of polynomials Pol in R[x_1,...,x_n] entirely supported on polynomials of degree at most d such that for all z in {0,1}^n, we have Pr_{P ~ Pol}[P(z) = f(z)] u003e= 1- epsilon. It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the epsilon-error probabilistic degree of the OR function is at most O(log n * log(1/epsilon)). Our first observation is that this can be improved to O{log (n atop u003c= log(1/epsilon))}, which is better for small values of epsilon.In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution Pol have the following special structure: P(x_1,...,x_n) = 1 - prod_{i in [t]} (1- L_i(x_1,...,x_n)), where each L_i(x_1,..., x_n) is a linear form in the variables x_1,...,x_n, i.e., the polynomial 1-P(bar{x}) is a product of affine forms. We show that the epsilon-error probabilistic degree of OR when restricted to polynomials of the above form is Omega(log (n over u003c= log(1/epsilon))/log^2 (log (n over u003c= log(1/epsilon))})), thus matching the above upper bound (up to polylogarithmic factors).
Year
DOI
Venue
2018
10.4230/LIPIcs.FSTTCS.2018.5
FSTTCS
DocType
Volume
ISSN
Journal
abs/1812.01982
In Proc. 38th IARCS Conf. on Foundations of Software Technology & Theoretical Computer Science (FSTTCS) (Ahmedabad, India, 10-14 December), volume 122 of LiPiCS, pages 5:1-5:12, 2018
Citations 
PageRank 
References 
0
0.34
0
Authors
4
Name
Order
Citations
PageRank
Siddharth Bhandari103.04
Prahladh Harsha237132.06
Tulasimohan Molli300.68
Srikanth Srinivasan413221.31