Abstract
We define the criticality of a boolean function f : {0,1}^n > {0,1} as the minimum real number lambda >= 1 such that Pr [DT_{depth}(fR_p) >= t] <= (p lambda)^t for all p in [0,1] and t in N, where R_p is the prandom restriction and DT_{depth} is decisiontree depth. Criticality is a useful parameter: it implies an O(2^((1 1/(2 lambda))n)) bound on the decisiontree size of f, as well as a 2^{Omega(k/lambda)} bound on Fourier weight of f on coefficients of size >= k.
In an unpublished manuscript [Rossmann, 2018], the author showed that a combination of Håstad's switching and multiswitching lemmas [Håstad, 1986; Håstad, 2014] implies that AC^0 circuits of depth d+1 and size s have criticality at most O(log s)^d. In the present paper, we establish a stronger O(1/d log s)^d bound for regular formulas: the class of AC^0 formulas in which all gates at any given depth have the same fanin. This result is based on
(i) a novel switching lemma for bounded size (unbounded width) DNF formulas, and
(ii) an extension of (i) which analyzes a canonical decision tree associated with an entire depthd formula.
As corollaries of our criticality bound, we obtain an improved #SAT algorithm and tight LinialMansourNisan Theorem for regular formulas, strengthening previous results for AC^0 circuits due to Impagliazzo, Matthews, Paturi [Impagliazzo et al., 2012] and Tal [Tal, 2017]. As a further corollary, we increase from o(log n /(log log n)) to o(log n) the number of quantifier alternations for which the QBFSAT (quantified boolean formula satisfiability) algorithm of Santhanam and Williams [Santhanam and Williams, 2014] beats exhaustive search.
BibTeX  Entry
@InProceedings{rossman:LIPIcs:2019:10823,
author = {Benjamin Rossman},
title = {{Criticality of Regular Formulas}},
booktitle = {34th Computational Complexity Conference (CCC 2019)},
pages = {1:11:28},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771160},
ISSN = {18688969},
year = {2019},
volume = {137},
editor = {Amir Shpilka},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10823},
URN = {urn:nbn:de:0030drops108230},
doi = {10.4230/LIPIcs.CCC.2019.1},
annote = {Keywords: AC^0 circuits, formulas, criticality}
}
Keywords: 

AC^0 circuits, formulas, criticality 
Collection: 

34th Computational Complexity Conference (CCC 2019) 
Issue Date: 

2019 
Date of publication: 

16.07.2019 