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AC0 Unpredictability

Published:17 March 2021Publication History
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Abstract

We prove that for every distribution D on n bits with Shannon entropy ≥ na, at most O(2da logd+1g)/γ5 of the bits Di can be predicted with advantage γ by an AC0 circuit of size g and depth D that is a function of all of the bits of D except Di. This answers a question by Meir and Wigderson, who proved a corresponding result for decision trees.

We also show that there are distributions D with entropy ≥ nO(1) such that any subset of O(n/ log n) bits of D on can be distinguished from uniform by a circuit of depth 2 and size poly(n). This separates the notions of predictability and distinguishability in this context.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 13, Issue 1
        March 2021
        143 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3447816
        Issue’s Table of Contents

        Copyright © 2021 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 17 March 2021
        • Accepted: 1 October 2020
        • Revised: 1 September 2020
        • Received: 1 November 2019
        Published in toct Volume 13, Issue 1

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