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Tight Correlation Bounds for Circuits Between AC0 and TC0

Author Vinayak M. Kumar

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Author Details

Vinayak M. Kumar
  • Department of Computer Science, University of Texas at Austin, TX, USA

Funding

  • Kumar, Vinayak M.: Supported by NSF Grant CCF-2008076 and a Simons Investigator Award (#409864, David Zuckerman).

Acknowledgements

The author thanks David Zuckerman and Chin Ho Lee for many valuable discussions, Xin Lyu for explaining his work in [Lyu, 2022], anonymous reviewers for valuable feedback and for pointing us to the construction presented in Theorem 54, and Jeffrey Champion, Shivam Gupta, Michael Jaber and Jiawei Li for helpful comments.

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Vinayak M. Kumar. Tight Correlation Bounds for Circuits Between AC0 and TC0. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 18:1-18:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CCC.2023.18

Abstract

We initiate the study of generalized AC⁰ circuits comprised of arbitrary unbounded fan-in gates which only need to be constant over inputs of Hamming weight ≥ k (up to negations of the input bits), which we denote GC⁰(k). The gate set of this class includes biased LTFs like the k-OR (outputs 1 iff ≥ k bits are 1) and k-AND (outputs 0 iff ≥ k bits are 0), and thus can be seen as an interpolation between AC⁰ and TC⁰. 
We establish a tight multi-switching lemma for GC⁰(k) circuits, which bounds the probability that several depth-2 GC⁰(k) circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-d size-s AC⁰ circuits lifts to depth-d size-s^{.99} GC⁰(.01 log s) circuits with no loss in parameters (other than hidden constants). 
Our result has the following applications:  
- Size-2^Ω(n^{1/d}) depth-d GC⁰(Ω(n^{1/d})) circuits do not correlate with parity (extending a result of Håstad (SICOMP, 2014)). 
- Size-n^Ω(log n) GC⁰(Ω(log² n)) circuits with n^{.249} arbitrary threshold gates or n^{.499} arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). 
- There is a seed length O((log m)^{d-1}log(m/ε)log log(m)) pseudorandom generator against size-m depth-d GC⁰(log m) circuits, matching the AC⁰ lower bound of Håstad up to a log log m factor (extending a result of Lyu (CCC, 2022)). 
- Size-m GC⁰(log m) circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • AC⁰
  • TC⁰
  • Switching Lemma
  • Lower Bounds
  • Correlation Bounds
  • Circuit Complexity

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