Improved average-case lower bounds for de Morgan formula size: Matching worst-case lower bound

Ilan Komargodski; Ilan Raz; Avishay Tal

doi:10.1137/15M1048045

Improved average-case lower bounds for de Morgan formula size: Matching worst-case lower bound

Ilan Komargodski, Ilan Raz, Avishay Tal

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

We give an explicit function h : {0; 1}ⁿ → {0; 1} such that every de Morgan formula of size n^3-o(1)=r² agrees with h on at most a fraction of 1/2 + 2^-ω(r) of the inputs. Our technical contributions include a theorem that shows that the "expected shrinkage" result of Hastad [SIAM J. Comput., 27 (1998), pp. 48-64] actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), using ideas of Impagliazzo, Meka, and Zuckerman [Proceedings of FOCS, 2012, pp. 111-119].

Original language	English
Pages (from-to)	37-57
Number of pages	21
Journal	SIAM Journal on Computing
Volume	46
Issue number	1
DOIs	https://doi.org/10.1137/15M1048045
State	Published - 2017
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.

Keywords

Average-case lower bound
Boolean formulas
Shrinkage

Access to Document

10.1137/15M1048045

Cite this

@article{80c697c36ee44600bc3c578b9287474b,

title = "Improved average-case lower bounds for de Morgan formula size: Matching worst-case lower bound",

abstract = "We give an explicit function h : {0; 1}n → {0; 1} such that every de Morgan formula of size n3-o(1)=r2 agrees with h on at most a fraction of 1/2 + 2-ω(r) of the inputs. Our technical contributions include a theorem that shows that the {"}expected shrinkage{"} result of Hastad [SIAM J. Comput., 27 (1998), pp. 48-64] actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), using ideas of Impagliazzo, Meka, and Zuckerman [Proceedings of FOCS, 2012, pp. 111-119].",

keywords = "Average-case lower bound, Boolean formulas, Shrinkage",

author = "Ilan Komargodski and Ilan Raz and Avishay Tal",

note = "Publisher Copyright: {\textcopyright} 2017 Society for Industrial and Applied Mathematics.",

year = "2017",

doi = "10.1137/15M1048045",

language = "אנגלית",

volume = "46",

pages = "37--57",

journal = "SIAM Journal on Computing",

issn = "0097-5397",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "1",

}

TY - JOUR

T1 - Improved average-case lower bounds for de Morgan formula size

T2 - Matching worst-case lower bound

AU - Komargodski, Ilan

AU - Raz, Ilan

AU - Tal, Avishay

PY - 2017

Y1 - 2017

N2 - We give an explicit function h : {0; 1}n → {0; 1} such that every de Morgan formula of size n3-o(1)=r2 agrees with h on at most a fraction of 1/2 + 2-ω(r) of the inputs. Our technical contributions include a theorem that shows that the "expected shrinkage" result of Hastad [SIAM J. Comput., 27 (1998), pp. 48-64] actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), using ideas of Impagliazzo, Meka, and Zuckerman [Proceedings of FOCS, 2012, pp. 111-119].

AB - We give an explicit function h : {0; 1}n → {0; 1} such that every de Morgan formula of size n3-o(1)=r2 agrees with h on at most a fraction of 1/2 + 2-ω(r) of the inputs. Our technical contributions include a theorem that shows that the "expected shrinkage" result of Hastad [SIAM J. Comput., 27 (1998), pp. 48-64] actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), using ideas of Impagliazzo, Meka, and Zuckerman [Proceedings of FOCS, 2012, pp. 111-119].

KW - Average-case lower bound

KW - Boolean formulas

KW - Shrinkage

UR - http://www.scopus.com/inward/record.url?scp=85014450460&partnerID=8YFLogxK

U2 - 10.1137/15M1048045

DO - 10.1137/15M1048045

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85014450460

SN - 0097-5397

VL - 46

SP - 37

EP - 57

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 1

ER -

Improved average-case lower bounds for de Morgan formula size: Matching worst-case lower bound

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this