{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:21:47Z","timestamp":1750220507868,"version":"3.41.0"},"reference-count":17,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T00:00:00Z","timestamp":1615939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"NSF","award":["NSF CCF award 1813930"],"award-info":[{"award-number":["NSF CCF award 1813930"]}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2021,3,31]]},"abstract":"<jats:p>\n            We prove that for every distribution\n            <jats:italic>D<\/jats:italic>\n            on\n            <jats:italic>n<\/jats:italic>\n            bits with Shannon entropy \u2265\n            <jats:italic>n<\/jats:italic>\n            \u2212\n            <jats:italic>a<\/jats:italic>\n            , at most\n            <jats:italic>O<\/jats:italic>\n            (2\n            <jats:sup>\n              <jats:italic>d<\/jats:italic>\n            <\/jats:sup>\n            <jats:italic>a<\/jats:italic>\n            log\n            <jats:sup>\n              <jats:italic>d<\/jats:italic>\n              +1\n            <\/jats:sup>\n            <jats:italic>g<\/jats:italic>\n            )\/\u03b3\n            <jats:sup>5<\/jats:sup>\n            of the bits\n            <jats:italic>D<\/jats:italic>\n            <jats:sub>\n              <jats:italic>i<\/jats:italic>\n            <\/jats:sub>\n            can be predicted with advantage \u03b3 by an AC\n            <jats:sup>0<\/jats:sup>\n            circuit of size\n            <jats:italic>g<\/jats:italic>\n            and depth\n            <jats:italic>D<\/jats:italic>\n            that is a function of all of the bits of\n            <jats:italic>D<\/jats:italic>\n            except\n            <jats:italic>D<\/jats:italic>\n            <jats:sub>\n              <jats:italic>i<\/jats:italic>\n            <\/jats:sub>\n            . This answers a question by Meir and Wigderson, who proved a corresponding result for decision trees.\n          <\/jats:p>\n          <jats:p>\n            We also show that there are distributions\n            <jats:italic>D<\/jats:italic>\n            with entropy \u2265\n            <jats:italic>n<\/jats:italic>\n            \u2212\n            <jats:italic>O<\/jats:italic>\n            (1) such that any subset of\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>n<\/jats:italic>\n            \/ log\n            <jats:italic>n<\/jats:italic>\n            ) bits of\n            <jats:italic>D<\/jats:italic>\n            on can be distinguished from uniform by a circuit of depth 2 and size poly(\n            <jats:italic>n<\/jats:italic>\n            ). This separates the notions of predictability and distinguishability in this context.\n          <\/jats:p>","DOI":"10.1145\/3442362","type":"journal-article","created":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T17:11:06Z","timestamp":1616001066000},"page":"1-8","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["AC0 Unpredictability"],"prefix":"10.1145","volume":"13","author":[{"given":"Emanuele","family":"Viola","sequence":"first","affiliation":[{"name":"Northeastern University, Boston, MA"}]}],"member":"320","published-online":{"date-parts":[[2021,3,17]]},"reference":[{"key":"e_1_2_1_1_1","first-page":"199","article-title":"Deterministic simulation of probabilistic constant-depth circuits","volume":"5","author":"Ajtai Miklos","year":"1989","unstructured":"Miklos Ajtai and Avi Wigderson . 1989 . Deterministic simulation of probabilistic constant-depth circuits . Advances in Computing Research: Randomness and Computation 5 (1989), 199 -- 223 . Miklos Ajtai and Avi Wigderson. 1989. Deterministic simulation of probabilistic constant-depth circuits. Advances in Computing Research: Randomness and Computation 5 (1989), 199--223.","journal-title":"Advances in Computing Research: Randomness and Computation"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-78381-9_9"},{"key":"e_1_2_1_3_1","series-title":"Wiley Series in Telecommunications and Signal Processing","volume-title":"Elements in Information Theory","author":"Cover Thomas","unstructured":"Thomas Cover and Joy Thomas . 2006. Elements in Information Theory . Wiley Series in Telecommunications and Signal Processing . Wiley InterScience . Thomas Cover and Joy Thomas. 2006. Elements in Information Theory. Wiley Series in Telecommunications and Signal Processing. 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Cambridge University Press."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2018.00094"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/120897432"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1963.10500830"},{"key":"e_1_2_1_11_1","first-page":"149","article-title":"Prediction from partial information and hindsight, with application to circuit lower bounds","volume":"24","author":"Meir Or","year":"2017","unstructured":"Or Meir and Avi Wigderson . 2017 . Prediction from partial information and hindsight, with application to circuit lower bounds . Electronic Colloquium on Computational Complexity (ECCC) 24 (2017), 149 . https:\/\/eccc.weizmann.ac.il\/report\/2017\/149 Or Meir and Avi Wigderson. 2017. Prediction from partial information and hindsight, with application to circuit lower bounds. Electronic Colloquium on Computational Complexity (ECCC) 24 (2017), 149. https:\/\/eccc.weizmann.ac.il\/report\/2017\/149","journal-title":"Electronic Colloquium on Computational Complexity (ECCC)"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539795280895"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1137\/080735096"},{"key":"e_1_2_1_14_1","first-page":"191","article-title":"Prediction from partial information and hindsight, an alternative proof","volume":"24","author":"Smal Alexander","year":"2017","unstructured":"Alexander Smal and Navid Talebanfard . 2017 . Prediction from partial information and hindsight, an alternative proof . Electronic Colloquium on Computational Complexity 24 (2017), 191 . https:\/\/eccc.weizmann.ac.il\/report\/2017\/191 Alexander Smal and Navid Talebanfard. 2017. Prediction from partial information and hindsight, an alternative proof. Electronic Colloquium on Computational Complexity 24 (2017), 191. https:\/\/eccc.weizmann.ac.il\/report\/2017\/191","journal-title":"Electronic Colloquium on Computational Complexity"},{"key":"e_1_2_1_15_1","series-title":"Lecture Notes in Computer Science","volume-title":"Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques","author":"Trevisan Luca","unstructured":"Luca Trevisan . 2004. A note on approximate counting for k-DNF . In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques . Lecture Notes in Computer Science , Vol. 3122 . Springer , 417--426. Luca Trevisan. 2004. A note on approximate counting for k-DNF. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. Lecture Notes in Computer Science, Vol. 3122. 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