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

Authors Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor C. Oliveira

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Valentine Kabanets
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Sajin Koroth
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Zhenjian Lu
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
  • Department of Computer Science, University of Warwick, UK
Dimitrios Myrisiotis
  • Department of Computing, Imperial College London, UK
Igor C. Oliveira
  • Department of Computer Science, University of Warwick, UK


We would like to thank Rocco Servedio for bringing to our attention the work by Kalai, Mansour, and Verbin [Adam Tauman Kalai et al., 2008], which is a central ingredient in the proof of Theorem 7. We also thank Mahdi Cheraghchi for several discussions on the analysis of Boolean circuits with a bottom layer of parity gates.

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Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, and Igor C. Oliveira. Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 15:1-15:41, Schloss Dagstuhl โ€“ Leibniz-Zentrum fรผr Informatik (2020)


The class ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [s]โˆ˜๐’ข consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class ๐’ข. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^{1.99}]โˆ˜๐’ข, for classes ๐’ข of functions with low communication complexity. Let R^(k)(๐’ข) be the maximum k-party number-on-forehead randomized communication complexity of a function in ๐’ข. Among other results, we show that: - The Generalized Inner Product function ๐–ฆ๐–จ๐–ฏ^k_n cannot be computed in ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [s]โˆ˜๐’ข on more than 1/2+ฮต fraction of inputs for s = o(nยฒ/{(kโ‹…4^kโ‹…R^(k)(๐’ข)โ‹…log (n/ฮต)โ‹…log(1/ฮต))ยฒ}). 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 ๐–ฆ๐–จ๐–ฏ^k_n against ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^{1.99}]โˆ˜๐–ฏ๐–ณ๐–ฅ^{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(โˆšsโ‹…R^(2)(๐’ข)โ‹…log(s/ฮต)โ‹…log(1/ฮต)) that ฮต-fools FORMULA[s]โˆ˜๐’ข. For the special case of FORMULA[s]โˆ˜๐–ซ๐–ณ๐–ฅ, i.e., size-s formulas with LTF (linear threshold function) gates at the bottom, we get the better seed length O(n^{1/2}โ‹…s^{1/4}โ‹…log(n)โ‹…log(n/ฮต)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n half-spaces in the regime where ฮต โ‰ค 1/n, complementing a recent result of [Ryan O'Donnell et al., 2019]. - There exists a randomized 2^{n-t}-time #SAT algorithm for ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [s]โˆ˜๐’ข, where t = ฮฉ(n/{โˆšsโ‹…logยฒ(s)โ‹…R^(2)(๐’ข)})^{1/2}. In particular, this implies a nontrivial #SAT algorithm for ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^1.99]โˆ˜๐–ซ๐–ณ๐–ฅ. - The Minimum Circuit Size Problem is not in ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^1.99]โˆ˜๐–ท๐–ฎ๐–ฑ; 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 ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^1.99]โˆ˜๐–ท๐–ฎ๐–ฑ can be PAC-learned in time 2^O(n/log n).

Subject Classification

ACM Subject Classification
  • Theory of computation โ†’ Pseudorandomness and derandomization
  • Theory of computation โ†’ Circuit complexity
  • Theory of computation โ†’ Communication complexity
  • Theory of computation โ†’ Complexity classes
  • de Morgan formulas
  • circuit lower bounds
  • satisfiability (SAT)
  • pseudorandom generators (PRGs)
  • learning
  • communication complexity
  • polynomial threshold functions (PTFs)
  • parities


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