License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2020.15
URN: urn:nbn:de:0030-drops-125673
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12567/
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Kabanets, Valentine ; Koroth, Sajin ; Lu, Zhenjian ; Myrisiotis, Dimitrios ; Oliveira, Igor C.

Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

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Abstract

The class ???????[s]∘? consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class ?. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^{1.99}]∘?, for classes ? of functions with low communication complexity. Let R^(k)(?) be the maximum k-party number-on-forehead randomized communication complexity of a function in ?. Among other results, we show that:
- The Generalized Inner Product function ???^k_n cannot be computed in ???????[s]∘? on more than 1/2+ε fraction of inputs for s = o(n²/{(k⋅4^k⋅R^(k)(?)⋅log (n/ε)⋅log(1/ε))²}). This significantly extends the lower bounds against bipartite formulas obtained by [Avishay Tal, 2017]. As a corollary, we get an average-case lower bound for ???^k_n against ???????[n^{1.99}]∘???^{k-1}, i.e., sub-quadratic-size de Morgan formulas with degree-(k-1) PTF (polynomial threshold function) gates at the bottom.
- There is a PRG of seed length n/2 + O(√s⋅R^(2)(?)⋅log(s/ε)⋅log(1/ε)) that ε-fools FORMULA[s]∘?. For the special case of FORMULA[s]∘???, i.e., size-s formulas with LTF (linear threshold function) gates at the bottom, we get the better seed length O(n^{1/2}⋅s^{1/4}⋅log(n)⋅log(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n half-spaces in the regime where ε ≤ 1/n, complementing a recent result of [Ryan O'Donnell et al., 2019].
- There exists a randomized 2^{n-t}-time #SAT algorithm for ???????[s]∘?, where t = Ω(n/{√s⋅log²(s)⋅R^(2)(?)})^{1/2}. In particular, this implies a nontrivial #SAT algorithm for ???????[n^1.99]∘???.
- The Minimum Circuit Size Problem is not in ???????[n^1.99]∘???; thereby making progress on hardness magnification, in connection with results from [Igor Carboni Oliveira et al., 2019; Lijie Chen et al., 2019]. On the algorithmic side, we show that the concept class ???????[n^1.99]∘??? can be PAC-learned in time 2^O(n/log n).

BibTeX - Entry

@InProceedings{kabanets_et_al:LIPIcs:2020:12567,
  author =	{Valentine Kabanets and Sajin Koroth and Zhenjian Lu and Dimitrios Myrisiotis and Igor C. Oliveira},
  title =	{{Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{15:1--15:41},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Shubhangi Saraf},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12567},
  URN =		{urn:nbn:de:0030-drops-125673},
  doi =		{10.4230/LIPIcs.CCC.2020.15},
  annote =	{Keywords: de Morgan formulas, circuit lower bounds, satisfiability (SAT), pseudorandom generators (PRGs), learning, communication complexity, polynomial threshold functions (PTFs), parities}
}

Keywords: de Morgan formulas, circuit lower bounds, satisfiability (SAT), pseudorandom generators (PRGs), learning, communication complexity, polynomial threshold functions (PTFs), parities
Collection: 35th Computational Complexity Conference (CCC 2020)
Issue Date: 2020
Date of publication: 17.07.2020


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