Circuit Lower Bounds for Average-Case MA

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9139)


Santhanam (2007) proved that \(\mathbf {MA}/1\) does not have circuits of size \(n^k\). We translate his result to the average-case setting by proving that there is a constant a such that for any k, there is a language in \(\mathrm {Avg}_{ }\mathbf {MA}\) that cannot be solved by circuits of size \(n^k\) on more than the \(1 - \frac{1}{n^a}\) fraction of inputs.

In order to get rid of the non-uniform advice, we supply the inputs with the probability threshold that we use to determine the acceptance. This technique was used by Pervyshev (2007) for proving a time hierarchy for heuristic computations.



The author is grateful to Edward A. Hirsch for bringing the problem to his attention, to Dmitry Itsykson and anonymous referees for their comments that significantly improved the (initially unreadable) presentation.


  1. [Ajt83]
    Ajtai, M.: \(\Sigma ^1_1\)-formulae on finite structures. Ann. Pure Appl. Logic 24, 1–48 (1983)MATHMathSciNetCrossRefGoogle Scholar
  2. [AW09]
    Aaronson, S., Wigderson, A.: Algebrization: a new barrier in complexity theory. TOCT 1(1), 1–54 (2009)CrossRefGoogle Scholar
  3. [Bar02]
    Barak, B.: A probabilistic-time hierarchy theorem for slightly non-uniform algorithms. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 194–208. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  4. [BFT98]
    Buhrman, H., Fortnow, L., Thierauf, T.: Nonrelativizing separations. In: IEEE Conference on Computational Complexity, pp. 8–12. IEEE Computer Society (1998)Google Scholar
  5. [Blu83]
    Blum, N.: A boolean function requiring 3n network size. Theor. Comput. Sci. 28(3), 337–345 (1983)CrossRefGoogle Scholar
  6. [BT06]
    Bogdanov, A., Trevisan, L.: Average-case complexity. Found. Trends Theor. Comput. Sci. 2(1), 1–106 (2006)MathSciNetCrossRefGoogle Scholar
  7. [Cai01]
    Cai, J.-Y.: \(S_{2} {\bf P} \subseteq {\bf ZPP}^{{\bf NP}}\). In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, pp. 620–629 (2001)Google Scholar
  8. [FS04]
    Fortnow , L., Santhanam, R.: Hierarchy theorems for probabilistic polynomial time. In: FOCS, pp. 316–324 (2004)Google Scholar
  9. [Hås86]
    Håstad, J.: Almost optimal lower bounds for small depth circuits. In: ACM STOC, pp. 6–20 (1986)Google Scholar
  10. [ILMR02]
    Iwama, K., Lachish, O., Morizumi, H., Raz, R.: An explicit lower bound of 5n - o(n) for boolean circuits. In: Diks, K., Rytter, W. (eds.) MFCF 2002. LNCS, vol. 2420. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  11. [IS14]
    Itsykson, D., Sokolov, D.: On fast non-deterministic algorithms and short heuristic proofs. Fundamenta Informaticae 132, 113–129 (2014)MATHMathSciNetGoogle Scholar
  12. [Its09]
    Itsykson, D.: Structural complexity of AvgBPP. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds.) CSR 2009. LNCS, vol. 5675, pp. 155–166. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  13. [Kan82]
    Kannan, R.: Circuit-size lower bounds and non-reducibility to sparse sets. Inf. Control 55(1), 40–56 (1982)MATHCrossRefGoogle Scholar
  14. [Per07]
    Pervyshev, K.: On heuristic time hierarchies. In: IEEE Conference of Computational Complexity, pp. 347–358 (2007)Google Scholar
  15. [Raz85]
    Razborov, A.: Lower bounds for the monotone complexity of some boolean functions. Doklady Akademii Nauk SSSR 281(4), 798–801 (1985)MathSciNetGoogle Scholar
  16. [San07]
    Santhanam, R.: Circuit lower bounds for Merlin-Arthur classes. In: ACM STOC, pp. 275–283 (2007)Google Scholar
  17. [Sha90]
    Shamir, A.: \({\bf IP = PSPACE}\). In: FOCS, pp. 11–15 (1990)Google Scholar
  18. [TV02]
    Trevisan, L., Vadhan, S.: Pseudorandomness and average-case complexity via uniform reductions. In: Proceedings of the 17th Annual IEEE Conference on Computational Complexity, pp. 129–138 (2002)Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Steklov Institute of Mathematics at St. PetersburgSt. PetersburgRussia

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