Summary: The Expressive Power of Voting Polynomials
James Aspnes y Richard Beigel z Merrick Furstx Steven Rudichx
October 14, 1993
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 low-degree 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 06520-8285.
zEmail: email@example.com. Supported in part by NSF grants CCR-8808949 and CCR-8958528.
xCarnegie-Mellon University, School of Computer Science, Pittsburgh, PA 15213-3890.
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 exponential-size lower bounds in a new way. This allows us to prove the new result