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

  • Ruiwen Chen
  • Igor C. Oliveira
  • Rahul Santhanam
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.


  1. 1.
    TCS Stack Exchange: How powerful is \(\sf ACC^0\) circuit class in average case? Accessed 27 Sept 2017
  2. 2.
    Ajtai, M.: \(\varSigma ^{1}_{1}\)-formulae on finite structures. Ann. Pure Appl. Logic 24(1), 1–48 (1983). MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arora, S., Barak, B.: Complexity Theory: A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Buresh-Oppenheim, J., Kabanets, V., Santhanam, R.: Uniform hardness amplification in NP via monotone codes. In: Electronic Colloquium on Computational Complexity (ECCC) TR06-154 (2006).
  5. 5.
    Carmosino, M.L., Impagliazzo, R., Kabanets, V., Kolokolova, A.: Learning algorithms from natural proofs. In: Conference on Computational Complexity (CCC), pp. 10:1–10:24 (2016).
  6. 6.
    Chen, R., Oliveira, I.C., Santhanam, R.: An average-case lower bound against ACC\(^0\). In: Electronic Colloquium on Computational Complexity (ECCC) TR17-173 (2017).
  7. 7.
    Fefferman, B., Shaltiel, R., Umans, C., Viola, E.: On beating the hybrid argument. Theory Comput. 9, 809–843 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17(1), 13–27 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goldwasser, S., Gutfreund, D., Healy, A., Kaufman, T., Rothblum, G.N.: Verifying and decoding in constant depth. In: Symposium on Theory of Computing (STOC), pp. 440–449 (2007).
  10. 10.
    Gutfreund, D., Rothblum, G.N.: The complexity of local list decoding. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX/RANDOM -2008. LNCS, vol. 5171, pp. 455–468. Springer, Heidelberg (2008). Google Scholar
  11. 11.
    Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Symposium on Theory of Computing (STOC), pp. 6–20 (1986).
  12. 12.
    Impagliazzo, R., Kabanets, V., Volkovich, I.: The power of natural properties as oracles. In: Electronic Colloquium on Computational Complexity (ECCC) TR17-023 (2017).
  13. 13.
    Nisan, N., Wigderson, A.: Hardness vs randomness. J. Comput. Syst. Sci. 49(2), 149–167 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Oliveira, I.C., Santhanam, R.: Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness. In: Computational Complexity Conference (CCC), pp. 18:1–18:49 (2017).
  15. 15.
    Oliveira, I.C., Santhanam, R.: Pseudodeterministic constructions in subexponential time. In: Symposium on Theory of Computing (STOC), pp. 665–677 (2017).
  16. 16.
    Razborov, A.A.: Lower bounds on the size of bounded-depth networks over the complete basis with logical addition. Math. Notes Acad. Sci. USSR 41(4), 333–338 (1987)zbMATHGoogle Scholar
  17. 17.
    Servedio, R., Tan, L.Y.: What circuit classes can be learned with non-trivial savings? In: Innovations in Theoretical Computer Science Conference (ITCS), pp. 1–23 (2017)Google Scholar
  18. 18.
    Shaltiel, R., Viola, E.: Hardness amplification proofs require majority. SIAM J. Comput. 39(7), 3122–3154 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In: Symposium on Theory of Computing (STOC), pp. 77–82 (1987).
  20. 20.
    Srinivasan, S.: On improved degree lower bounds for polynomial approximation. In: Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp. 201–212 (2013).
  21. 21.
    Sudan, M., Trevisan, L., Vadhan, S.P.: Pseudorandom generators without the XOR lemma. J. Comput. Syst. Sci. 62(2), 236–266 (2001). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Williams, R.: Nonuniform ACC circuit lower bounds. J. ACM 61(1), 2:1–2:32 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Williams, R.: Natural proofs versus derandomization. SIAM J. Comput. 45(2), 497–529 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yao, A.C.: Separating the polynomial-time hierarchy by oracles (preliminary version). In: Symposium on Foundations of Computer Science (FOCS), pp. 1–10 (1985).

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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