 
Summary: The Expressive Power of Voting Polynomials
James Aspnes y Richard Beigel z Merrick Furstx Steven Rudichx
October 14, 1993
Abstract
We consider the problem of approximating a Boolean function f : f0;1gn ! f0;1g by the
sign of an integer polynomial p of degree k. For us, a polynomialp(x) predicts the value of f(x)
if, whenever p(x) 0, f(x) = 1, and whenever p(x) < 0, f(x) = 0. A lowdegree polynomial
p is a good approximator for f if it predicts f at almost all points. Given a positive integer
k, and a Boolean function f, we ask, \how good is the best degree k approximation to f?"
We introduce a new lower bound technique which applies to any Boolean function. We show
that the lower bound technique yields tight bounds in the case f is parity. Minsky and Papert
10] proved that a perceptron can not compute parity; our bounds indicate exactly how well
Yale University, Dept. of Computer Science, P.O. Box 208285, New Haven CT 065208285.
yEmail: aspnesjames@cs.yale.edu.
zEmail: beigelrichard@cs.yale.edu. Supported in part by NSF grants CCR8808949 and CCR8958528.
xCarnegieMellon University, School of Computer Science, Pittsburgh, PA 152133890.
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a perceptron can approximate it. As a consequence, we are able to give the rst correct proof
that, for a random oracle A, PPA is properly contained in PSPACEA. We are also able to prove
the old AC0 exponentialsize lower bounds in a new way. This allows us to prove the new result
