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Uniqueness of Optimal Mod 3 Circuits for Parity

Authors: Frederic Green and Amitabha Roy

Published in: Dagstuhl Seminar Proceedings, Volume 7411, Algebraic Methods in Computational Complexity (2008)

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.

Cite as

Frederic Green and Amitabha Roy. Uniqueness of Optimal Mod 3 Circuits for Parity. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 7411, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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  author =	{Green, Frederic and Roy, Amitabha},
  title =	{{Uniqueness of Optimal Mod 3 Circuits  for Parity}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--15},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{7411},
  editor =	{Manindra Agrawal and Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-13059},
  doi =		{10.4230/DagSemProc.07411.7},
  annote =	{Keywords: Circuit complexity, correlations, exponential sums}
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