Affiliation:
1. Institute of Mathematics of ASCR, Prague, Praha, Czech Republic
Abstract
For a Boolean function
f
: {0, 1}
n
→ {0, 1}, let
fˆ
be the unique multilinear polynomial such that
f
(
x
) =
fˆ
(
x
) holds for every
x
ˆ {0, 1}
n
. We show that, assuming VP ≠ VNP, there exists a polynomial-time computable
f
such that
fˆ
requires superpolynomial arithmetic circuits. In fact, this
f
can be taken as a monotone 2-CNF, or a product of affine functions.
This holds over any field. To prove the results in characteristic 2, we design new VNP-complete families in this characteristic. This includes the polynomial EC
n
counting edge covers in a graph and the polynomial mclique
n
counting cliques in a graph with deleted perfect matching. They both correspond to polynomial-time decidable problems, a phenomenon previously encountered only in characteristic ≠ 2.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
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