Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs

Aryeh Grinberg, Ronen Shaltiel, Emanuele Viola

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We study how well can q-query decision trees distinguish between the following two distributions: (i) R = (R 1 ,⋯, R N ) that are i.i.d. indicator random variables, (ii) X = (R|R ∈ A) where A is an event s.t. Pr[R∈A] ≥ 2 -a . We prove two lemmas: • Forbidden-set lemma: There exists B ⊆ [N] of size poly(a, q, 1/η) such that q-query trees that do not query variables in B cannot distinguish X from R with advantage η. • Fixed-set lemma: There exists B ⊆ [N] of size poly(a, q, 1/η) and υ ∈ {0, 1} B such that q-query trees do not distinguish (X|X B = v) from (R|R B = v) with advantage η. The first can be seen as an extension of past work by Edmonds, Impagliazzo, Rudich and Sgall (Computational Complexity 2001), Raz (SICOMP 1998), and Shaltiel and Viola (SICOMP 2010) to adaptive decision trees. It is independent of recent work by Meir and Wigderson (ECCC 2017) bounding the number of i ∈ [N] for which there exists a q-query tree that predicts Xi from the other bits. We use the second, fixed-set lemma to prove lower bounds on black-box proofs for hardness amplification that amplify hardness from δ to 1/2 - ϵ. Specifically: • Reductions must make q = Ω(log(1/δ)/ϵ 2 ) queries, implying a "size loss factor" of q. We also prove the lower bound q = Ω(log(1/δ)/ϵ) for "error-less" hardness amplification proofs, and for direct-product lemmas. These bounds are tight. • Reductions can be used to compute Majority on Ω(1/ϵ) bits, implying that black box proofs cannot amplify hardness of functions that are hard against constant depth circuits (unless they are allowed to use Majority gates). Both items extend to pseudorandom-generator constructions. These results prove 15-year-old conjectures by Viola, and improve on three incomparable previous works (Shaltiel and Viola, SICOMP 2010; Gutfreund and Rothblum, RANDOM 2008; Artemenko and Shaltiel, Computational Complexity 2014).

Original languageEnglish
Title of host publicationProceedings - 59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018
EditorsMikkel Thorup
PublisherIEEE Computer Society
Pages956-966
Number of pages11
ISBN (Electronic)9781538642306
DOIs
StatePublished - 30 Nov 2018
Event59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018 - Paris, France
Duration: 7 Oct 20189 Oct 2018

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2018-October
ISSN (Print)0272-5428

Conference

Conference59th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2018
Country/TerritoryFrance
CityParis
Period7/10/189/10/18

Bibliographical note

Publisher Copyright:
© 2018 IEEE.

Keywords

  • Black-box impossibility
  • Decision trees
  • Hardness amplification

ASJC Scopus subject areas

  • Computer Science (all)

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