Abstract
The class 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢 consists of Boolean functions computable by sizes de Morgan formulas whose leaves are any Boolean functions from a class 𝒢. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^{1.99}]∘𝒢, for classes 𝒢 of functions with low communication complexity. Let R^(k)(𝒢) be the maximum kparty numberonforehead randomized communication complexity of a function in 𝒢. Among other results, we show that:
 The Generalized Inner Product function 𝖦𝖨𝖯^k_n cannot be computed in 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢 on more than 1/2+ε fraction of inputs for s = o(n²/{(k⋅4^k⋅R^(k)(𝒢)⋅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 averagecase lower bound for 𝖦𝖨𝖯^k_n against 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^{1.99}]∘𝖯𝖳𝖥^{k1}, i.e., subquadraticsize de Morgan formulas with degree(k1) PTF (polynomial threshold function) gates at the bottom.
 There is a PRG of seed length n/2 + O(√s⋅R^(2)(𝒢)⋅log(s/ε)⋅log(1/ε)) that εfools FORMULA[s]∘𝒢. For the special case of FORMULA[s]∘𝖫𝖳𝖥, i.e., sizes 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 nontrivial PRG (with seed length o(n)) for intersections of n halfspaces in the regime where ε ≤ 1/n, complementing a recent result of [Ryan O'Donnell et al., 2019].
 There exists a randomized 2^{nt}time #SAT algorithm for 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢, where t = Ω(n/{√s⋅log²(s)⋅R^(2)(𝒢)})^{1/2}. In particular, this implies a nontrivial #SAT algorithm for 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖫𝖳𝖥.
 The Minimum Circuit Size Problem is not in 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖷𝖮𝖱; 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 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖷𝖮𝖱 can be PAClearned 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 LowCommunication Leaf Gates}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {15:115:41},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771566},
ISSN = {18688969},
year = {2020},
volume = {169},
editor = {Shubhangi Saraf},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12567},
URN = {urn:nbn:de:0030drops125673},
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 