On TC0 Lower Bounds for the Permanent

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