Advertisement

Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth

  • Maurice Jansen
  • Rahul Santhanam
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φ n }, where Φ n has size at most p(n) and depth O(1), such that Φ n computes the n×n permanent. A circuit family {Φ n } is succinct if there exists a nonuniform Boolean circuit family {C n } with O(logn) many inputs and size n o(1) such that that C n can correctly answer direct connection language queries about Φ n - succinctness is a relaxation of uniformity.

To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this we obtain the lower bound by explicitly constructing a hitting set against arithmetic circuits in the polynomial hierarchy.

Keywords

Boolean Function Complexity Class Integer Sequence Arithmetic Circuit Polynomial Size 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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.
    Agrawal, M., Kayal, N., Saxena, N.: Primes is in P. Ann. of Math 2, 781–793 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Agrawal, M., Vinay, V.: Arithmetic circuits: A chasm at depth four. In: Proc. 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 67–75 (2008)Google Scholar
  4. 4.
    Allender, E.: The permanent requires large uniform threshold circuits. Chicago Journal of Theoretical Computer Science, article 7 (1999)Google Scholar
  5. 5.
    Barrington, D.M., Immerman, N., Straubing, H.: On uniformity within NC1. J. Comp. Sys. Sci. 41, 274–306 (1990)CrossRefzbMATHGoogle Scholar
  6. 6.
    Baur, W., Strassen, V.: The complexity of partial derivatives. Theor. Comp. Sci. 22, 317–330 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bürgisser, P.: On defining integers and proving arithmetic circuit lower bounds. Computational Complexity 18, 81–103 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    DeMillo, R., Lipton, R.: A probabilistic remark on algebraic program testing. Inf. Proc. Lett. 7, 193–195 (1978)CrossRefzbMATHGoogle Scholar
  9. 9.
    Grigoriev, D., Karpinski, M.: An exponential lower bound for depth 3 arithmetic circuits. In: Proc. 13th Annual ACM Symposium on the Theory of Computing, pp. 577–582 (1998)Google Scholar
  10. 10.
    Heintz, J., Schnorr, C.: Testing polynomials which are easy to compute (extended abstract). In: Proc. 12th Annual ACM Symposium on the Theory of Computing, pp. 262–272 (1980)Google Scholar
  11. 11.
    Hesse, W., Allender, E., Barrington, D.: Uniform constant-depth threshold circuits for division and iterated multiplication. J. Comp. Sys. Sci. 64(4), 695–716 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ibarra, O., Moran, S.: Probabilistic algorithms for deciding equivalence of straight-line programs. J. Assn. Comp. Mach. 30, 217–228 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jansen, M., Santhanam, R.: Marginal hitting sets imply super-polynomial lower bounds for permanent (2011) (manuscript)Google Scholar
  14. 14.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity testing means proving circuit lower bounds. Computational Complexity 13(1-2), 1–44 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koiran, P.: Shallow circuits with high powered inputs. In: Proc. 2nd Symp. on Innovations in Computer Science (2010)Google Scholar
  16. 16.
    Koiran, P., Perifel, S.: Interpolation in Valiant’s theory (2007) (to appear)Google Scholar
  17. 17.
    Koiran, P., Perifel, S.: A superpolynomial lower bound on the size of uniform non-constant-depth threshold circuits for the permanent. In: Proc. 24th Annual IEEE Conference on Computational Complexity (2009)Google Scholar
  18. 18.
    Raz, R.: Elusive functions and lower bounds for arithmetic circuits. Theory of Computing 6 (2010)Google Scholar
  19. 19.
    Raz, R., Yehudayoff, A.: Lower bounds and separations for constant depth multilinear circuits. Computational Complexity 18(2) (2009)Google Scholar
  20. 20.
    Schwartz, J.: Fast probabilistic algorithms for polynomial identities. J. Assn. Comp. Mach. 27, 701–717 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shpilka, A., Wigderson, A.: Depth-3 arithmetic formulae over fields of characteristic zero. Journal of Computational Complexity 10(1), 1–27 (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20, 865–877 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Torán, J.: Complexity classes defined by counting quantifiers. J. Assn. Comp. Mach. 38(3), 753–774 (1991)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Valiant, L.: The complexity of computing the permanent. Theor. Comp. Sci. 8, 189–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Vollmer, H.: Introduction to Circuit Complexity. A uniform approach. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  26. 26.
    Wagner, K.: The complexity of combinatorial problems with succinct input representation. Acta Informatica 23, 325–356 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zankó, V.: #P-completeness via many-one reductions. International Journal of Foundations of Computer Science 2, 77–82 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Ng, K.W. (ed.) EUROSAM 1979 and ISSAC 1979. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Maurice Jansen
    • 1
  • Rahul Santhanam
    • 1
  1. 1.School of InformaticsThe University of EdinburghScotland, UK

Personalised recommendations