On TC0 Lower Bounds for the Permanent

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

Abstract

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) < 2n. 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 nO(1).

     
  3. 3

    Depth O(1), no(1) bits of non-uniformity, and size nO(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 no(1) bits of non-uniformity, or the permanent does not have polynomial-size general circuits.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

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