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

14 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

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10.1137/15M1048045

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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].",

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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].

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

Abstract

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Keywords

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