On the Correlation of Parity and Small-Depth Circuits

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Previous article Next article On the Correlation of Parity and Small-Depth CircuitsJohan HåstadJohan Håstadhttps://doi.org/10.1137/120897432PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe prove that the correlation of a depth-$d$ unbounded fanin circuit of size $S$ with parity of $n$ variables is at most $2^{-\Omega(n/(\log S)^{d-1})}$.1. M. Ajtai on finite structures , Ann. Pure Appl. Logic , 24 ( 1983 ), pp. 1 -- 48 . CrossrefISIGoogle Scholar2. P. Beame, R. Impagliazzo and S. Srinivasan , Approximating $AC^0$ by small height decision trees and a deterministic algorithm for $\#AC^0 SAT$ , in Proceedings of the IEEE Conference on Computational Complexity , 2012 , pp. 117 -- 125 . Google Scholar3. J.-Y. Cai , With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy , J. Comput. System Sci. , 38 ( 1989 ), pp. 68 -- 85 . CrossrefISIGoogle Scholar4. M. Furst, J. B. Saxe and M. 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Yao, Separating the polynomial-time hierarchy by oracles, in Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, FOCS '85, 1985, pp. 1--10.Google ScholarKeywordscircuits complexitysmall-depth circuitsparityswitching lemma Previous article Next article FiguresRelatedReferencesCited ByDetails Expander-Based Cryptography Meets Natural Proofscomputational complexity, Vol. 31, No. 1 | 16 March 2022 Cross Ref Oracle Separation of BQP and PHJournal of the ACM | 4 May 2022 Cross Ref Correction to: Unconditionally Secure Computation Against Low-Complexity LeakageJournal of Cryptology, Vol. 35, No. 2 | 24 January 2022 Cross Ref Fooling Constant-Depth Threshold Circuits (Extended Abstract)2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) | 1 Feb 2022 Cross Ref Tradeoffs for small-depth Frege proofs2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) | 1 Feb 2022 Cross Ref Unconditionally Secure Computation Against Low-Complexity LeakageJournal of Cryptology, Vol. 34, No. 4 | 9 August 2021 Cross Ref AC0 UnpredictabilityACM Transactions on Computation Theory, Vol. 13, No. 1 | 1 Mar 2021 Cross Ref Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizationsScience, Vol. 367, No. 6481 | 28 Feb 2020 Cross Ref Sampling Lower Bounds: Boolean Average-Case and PermutationsEmanuele ViolaSIAM Journal on Computing, Vol. 49, No. 1 | 18 February 2020AbstractPDF (479 KB)Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuitsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing | 23 June 2019 Cross Ref On polynomial approximations to AC0Random Structures & Algorithms, Vol. 54, No. 2 | 12 July 2018 Cross Ref Unconditionally Secure Computation Against Low-Complexity LeakageAdvances in Cryptology – CRYPTO 2019 | 1 August 2019 Cross Ref Gate elimination: Circuit size lower bounds and #SAT upper boundsTheoretical Computer Science, Vol. 719 | 1 Apr 2018 Cross Ref Bounded Independence Plus Noise Fools ProductsElad Haramaty, Chin Ho Lee, and Emanuele ViolaSIAM Journal on Computing, Vol. 47, No. 2 | 12 April 2018AbstractPDF (574 KB)An Average-Case Depth Hierarchy Theorem for Boolean CircuitsJournal of the ACM, Vol. 64, No. 5 | 15 Oct 2017 Cross Ref Fourier Concentration from Shrinkagecomputational complexity, Vol. 26, No. 1 | 12 May 2016 Cross Ref An Average-Case Depth Hierarchy Theorem for Higher Depth2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) | 1 Oct 2016 Cross Ref Super-linear gate and super-quadratic wire lower bounds for depth-two and depth-three threshold circuitsProceedings of the forty-eighth annual ACM symposium on Theory of Computing | 19 June 2016 Cross Ref Fine-Grained CryptographyAdvances in Cryptology – CRYPTO 2016 | 21 July 2016 Cross Ref Bounded Indistinguishability and the Complexity of Recovering SecretsAdvances in Cryptology – CRYPTO 2016 | 21 July 2016 Cross Ref An Average-Case Depth Hierarchy Theorem for Boolean Circuits2015 IEEE 56th Annual Symposium on Foundations of Computer Science | 1 Oct 2015 Cross Ref Volume 43, Issue 5| 2014SIAM Journal on Computing1513-1830 History Submitted:11 November 2012Accepted:24 June 2014Published online:25 September 2014 Information© 2014, Society for Industrial and Applied MathematicsKeywordscircuits complexitysmall-depth circuitsparityswitching lemmaMSC codes68Q17PDF Download Article & Publication DataArticle DOI:10.1137/120897432Article page range:pp. 1699-1708ISSN (print):0097-5397ISSN (online):1095-7111Publisher:Society for Industrial and Applied Mathematics