Lower Bounds against Weakly Uniform Circuits

  • Ruiwen Chen
  • Valentine Kabanets
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


A family of Boolean circuits \(\{C_n\}_{n\geqslant 0}\) is called γ(n)-weakly uniform if there is a polynomial-time algorithm for deciding the direct-connection language of every C n , given advice of size γ(n). This is a relaxation of the usual notion of uniformity, which allows one to interpolate between complete uniformity (when γ(n) = 0) and complete non-uniformity (when γ(n) > |C n |). Weak uniformity is essentially equivalent to succinctness introduced by Jansen and Santhanam [12].

Our main result is that Permanent is not computable by polynomial-size n o(1)-weakly uniform TC 0 circuits. This strengthens the results by Allender [2] (for uniform TC 0) and by Jansen and Santhanam [12] (for weakly uniform arithmetic circuits of constant depth). Our approach is quite general, and can be used to extend to the “weakly uniform” setting all currently known circuit lower bounds proved for the “uniform” setting. For example, we show that Permanent is not computable by polynomial-size (logn) O(1)-weakly uniform threshold circuits of depth o(loglogn), generalizing the result by Koiran and Perifel [16].


advice complexity classes alternating Turing machines counting hierarchy permanent succinct circuits threshold circuits uniform circuit lower bounds weakly uniform circuits 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrawal, M.: Proving Lower Bounds Via Pseudo-random Generators. In: Sarukkai, S., Sen, S. (eds.) FSTTCS 2005. LNCS, vol. 3821, pp. 92–105. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Allender, E.: The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science (1999)Google Scholar
  3. 3.
    Allender, E., Gore, V.: A uniform circuit lower bound for the permanent. SIAM Journal on Computing 23(5), 1026–1049 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Arora, S., Barak, B.: Complexity theory: a modern approach. CUP, NY (2009)zbMATHCrossRefGoogle Scholar
  5. 5.
    Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC 1. JCSS 41, 274–306 (1990)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, R., Kabanets, V.: Lower bounds against weakly uniform circuits. In: ECCC, vol. 19, p. 7 (2012)Google Scholar
  7. 7.
    Chandra, A., Kozen, D., Stockmeyer, L.: Alternation. JACM 28(1), 114 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17(1), 13–27 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Håstad, J.: Almost optimal lower bounds for small depth circuits. In: STOC 1986 (1986)Google Scholar
  10. 10.
    Heintz, J., Schnorr, C.-P.: Testing polynomials which are easy to compute. L’Enseignement Mathématique 30, 237–254 (1982)MathSciNetGoogle Scholar
  11. 11.
    Iwama, K., Morizumi, H.: An Explicit Lower Bound of 5n-o(n) for Boolean Circuits. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 353–364. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity 13(1–2), 1–46 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kannan, R.: Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control 55, 40–56 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Karp, R.M., Lipton, R.J.: Turing machines that take advice. L’Enseignement Mathématique 28(3-4), 191–209 (1982)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Koiran, P., Perifel, S.: A superpolynomial lower bound on the size of uniform non-constant-depth threshold circuits for the permanent. In: CCC (2009)Google Scholar
  17. 17.
    Lachish, O., Raz, R.: Explicit lower bound of 4.5n − o(n) for boolean circuits. In: Proc. of the Thirty-Third ACM Symp. on Theory of Computing, pp. 399–408 (2001)Google Scholar
  18. 18.
    Lupanov, O.B.: On the synthesis of switching circuits. Doklady Akademii Nauk SSSR 119(1), 23–26 (1958); English translation in Soviet Mathematics DokladyMathSciNetzbMATHGoogle Scholar
  19. 19.
    Parberry, I., Schnitger, G.: Parallel computation with threshold functions. In: Proc. of the First IEEE Conf. on Structure in Complexity Theory, pp. 272–290 (1986)Google Scholar
  20. 20.
    Razborov, A.A.: Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes 41, 333–338 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Razborov, A.A., Rudich, S.: Natural proofs. JCSS 55, 24–35 (1997)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ruzzo, W.L.: On uniform circuit complexity. JCSS 22(3), 365–383 (1981)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Shannon, C.E.: The synthesis of two-terminal switching circuits. Bell System Technical Journal 28(1), 59–98 (1949)MathSciNetGoogle Scholar
  24. 24.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: Proc. of the Nineteenth ACM STOC, pp. 77–82 (1987)Google Scholar
  25. 25.
    Torán, J.: Complexity classes defined by counting quantifiers. JACM 38, 752 (1991)CrossRefGoogle Scholar
  26. 26.
    Valiant, L.: The complexity of computing the permanent. TCS 8, 189–201 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Wagner, K.W.: The complexity of combinatorial problems with succinct input representation. Acta Informatica 23, 325–356 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Williams, R.: Non-uniform ACC circuit lower bounds. In: CCC (2011)Google Scholar
  29. 29.
    Yao, A.C.: Separating the polynomial-time hierarchy by oracles. In: FOCS (1985)Google Scholar
  30. 30.
    Zak, S.: A Turing machine hierarchy. TCS 26, 327–333 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Zanko, V.: #P-Completeness via Many-One Reductions. IJFCS 1, 77 (1991)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ruiwen Chen
    • 1
  • Valentine Kabanets
    • 1
  1. 1.School of Computing ScienceSimon Fraser UniversityBurnabyCanada

Personalised recommendations