On TC0 Lower Bounds for the Permanent

  • Jeff Kinne
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


In this paper we consider the problem of proving lower bounds for the permanent. An ongoing line of research has shown super-polynomial lower bounds for slightly-non-uniform small-depth threshold and arithmetic circuits [1,2,3,4]. We prove a new parameterized lower bound that includes each of the previous results as sub-cases. Our main result implies that the permanent does not have Boolean threshold circuits of the following kinds.
  1. 1

    Depth O(1), poly − log(n) bits of non-uniformity, and size s(n) such that for all constants c, s (c)(n) < 2 n . The size s must satisfy another technical condition that is true of functions normally dealt with (such as compositions of polynomials, logarithms, and exponentials).

  2. 2

    Depth o(loglogn), poly − log(n) bits of non-uniformity, and size n O(1).

  3. 3

    Depth O(1), n o(1) bits of non-uniformity, and size n O(1).


Our proof yields a new “either or” hardness result. One instantiation is that either NP does not have polynomial-size constant-depth threshold circuits that use n o(1) bits of non-uniformity, or the permanent does not have polynomial-size general circuits.


Turing Machine Arithmetic Circuit Input Length Majority Gate Threshold Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Allender, E.: The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science (1999)Google Scholar
  2. 2.
    Koiran, P., Perifel, S.: A superpolynomial lower bound on the size of uniform non-constant-depth threshold circuits for the permanent. In: Proceedings of the IEEE Conference on Computational Complexity (CCC), pp. 35–40 (2009)Google Scholar
  3. 3.
    Jansen, M., Santhanam, R.: Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 724–735. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Jansen, M., Santhanam, R.: Marginal hitting sets imply super-polynomial lower bounds for permanent. In: Innovations in Theoretical Computer Science (2012)Google Scholar
  5. 5.
    Nisan, N., Wigderson, A.: Hardness vs. randomness. Journal of Computer and System Sciences 49(2), 149–167 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity 3, 307–318 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 220–229 (1997)Google Scholar
  8. 8.
    Buhrman, H., Fortnow, L., Thierauf, T.: Nonrelativizing separations. In: Proceedings of the IEEE Conference on Computational Complexity (CCC), pp. 8–12 (1998)Google Scholar
  9. 9.
    Miltersen, P.B., Vinodchandran, N.V., Watanabe, O.: Super-Polynomial Versus Half-Exponential Circuit Size in the Exponential Hierarchy. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 210–220. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. 10.
    Williams, R.: Non-uniform ACC circuit lower bounds. In: Proceedings of the IEEE Conference on Computational Complexity (CCC), pp. 115–125 (2011)Google Scholar
  11. 11.
    Allender, E.: Circuit complexity before the dawn of the new millennium. In: Proceedings of the Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp. 1–18 (1996)Google Scholar
  12. 12.
    Kinne, J., van Melkebeek, D., Shaltiel, R.: Pseudorandom Generators and Typically-Correct Derandomization. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX and RANDOM 2009. LNCS, vol. 5687, pp. 574–587. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Dvir, Z., Shpilka, A., Yehudayoff, A.: Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM Journal on Computing 39(4), 1279–1293 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Zanko, V.: #P-completeness via many-one reductions. International Journal of Foundations of Computer Science 2, 77–82 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing 20, 865–877 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kannan, R.: Circuit-size lower bounds and nonreducibility to sparse sets. Information and Control 55, 40–56 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity 13(1/2), 1–46 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Aaronson, S., van Melkebeek, D.: A note on circuit lower bounds from derandomization. Electronic Colloquium on Computational Complexity 17 (2010)Google Scholar
  19. 19.
    Vinodchandran, N.V.: A note on the circuit complexity of PP. Theoretical Computer Science 347(1-2), 415–418 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Impagliazzo, R., Kabanets, V., Wigderson, A.: In search of an easy witness: exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences 65(4), 672–694 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chen, R., Kabanets, V.: Lower bounds against weakly uniform circuits. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 408–419. Springer, Heidelberg (2012)Google Scholar
  22. 22.
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8, 189–201 (1979)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jeff Kinne
    • 1
  1. 1.Indiana State UniversityUSA

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