eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2008-01-15
1
15
10.4230/DagSemProc.07411.7
article
Uniqueness of Optimal Mod 3 Circuits for Parity
Green, Frederic
Roy, Amitabha
We prove that the quadratic polynomials modulo $3$
with the largest correlation with parity are unique up to
permutation of variables and constant factors. As a consequence of
our result, we completely characterize the smallest
MAJ~$circ mbox{MOD}_3 circ {
m AND}_2$ circuits that compute parity, where a
MAJ~$circ mbox{MOD}_3 circ {
m AND}_2$ circuit is one that has a
majority gate as output, a middle layer of MOD$_3$ gates and a
bottom layer of AND gates of fan-in $2$. We
also prove that the sub-optimal circuits exhibit a stepped behavior:
any sub-optimal circuits of this class that compute parity
must have size at least a factor of $frac{2}{sqrt{3}}$ times the
optimal size. This verifies, for the special case of $m=3$,
two conjectures made
by Due~{n}ez, Miller, Roy and Straubing (Journal of Number Theory, 2006) for general MAJ~$circ mathrm{MOD}_m circ
{
m AND}_2$ circuits for any odd $m$. The correlation
and circuit bounds are obtained by studying the associated
exponential sums, based on some of the techniques developed
by Green (JCSS, 2004). We regard this as a step towards
obtaining tighter bounds both for the $m
ot = 3$ quadratic
case as well as for
higher degrees.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol07411/DagSemProc.07411.7/DagSemProc.07411.7.pdf
Circuit complexity
correlations
exponential sums