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DOI: 10.4230/LIPIcs.FSTTCS.2018.5
URN: urn:nbn:de:0030-drops-99044
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9904/
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Bhandari, Siddharth ; Harsha, Prahladh ; Molli, Tulasimohan ; Srinivasan, Srikanth

On the Probabilistic Degree of OR over the Reals

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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 -> {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)] >= 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 <= 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 <= log(1/epsilon))/log^2 (log (n over <= log(1/epsilon))})), thus matching the above upper bound (up to polylogarithmic factors).

BibTeX - Entry

@InProceedings{bhandari_et_al:LIPIcs:2018:9904,
  author =	{Siddharth Bhandari and Prahladh Harsha and Tulasimohan Molli and Srikanth Srinivasan},
  title =	{{On the Probabilistic Degree of OR over the Reals}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software  Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{5:1--5:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Sumit Ganguly and Paritosh Pandya},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9904},
  URN =		{urn:nbn:de:0030-drops-99044},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.5},
  annote =	{Keywords: Polynomials over reals, probabilistic polynomials, probabilistic degree, OR polynomial}
}

Keywords: Polynomials over reals, probabilistic polynomials, probabilistic degree, OR polynomial
Seminar: 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)
Issue Date: 2018
Date of publication: 23.11.2018


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