An Average-Case Lower Bound Against \(\mathsf {ACC}^0\)

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


In a seminal work, Williams [22] showed that \(\mathsf {NEXP}\) (non-deterministic exponential time) does not have polynomial-size \(\mathsf {ACC}^0\) circuits. Williams’ technique inherently gives a worst-case lower bound, and until now, no average-case version of his result was known. We show that there is a language L in \(\mathsf {NEXP}\) and a function \(\varepsilon (n) = 1/\log (n)^{\omega (1)}\) such that no sequence of polynomial size \(\mathsf {ACC}^0\) circuits solves L on more than a \(1/2+\varepsilon (n)\) fraction of inputs of length n for all large enough n. Complementing this result, we give a nontrivial pseudo-random generator against polynomial-size \(\mathsf {AC}^0[6]\) circuits. We also show that learning algorithms for quasi-polynomial size \(\mathsf {ACC}^0\) circuits running in time \(2^n/n^\omega (1)\) imply lower bounds for the randomised exponential time classes \(\mathsf {RE}\) (randomized time \(2^{O(n)}\) with one-sided error) and \(\mathsf {ZPE}/1\) (zero-error randomized time \(2^{O(n)}\) with 1 bit of advice) against polynomial size \(\mathsf {ACC}^0\) circuits. This strengthens results of Oliveira and Santhanam [15].


Circuit lower bounds Average-case complexity Pseudorandomness Learning and natural properties 



We would like to thank Marco Carmosino for posing the question of proving average-case hardness against \(\mathsf {ACC}^0\), and for useful discussions. This work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agrement no. 615075.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ruiwen Chen
    • 1
  • Igor C. Oliveira
    • 1
  • Rahul Santhanam
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK

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