Are all distributions easy?

Complexity theory typically studies the complexity of computing a function h(x) : {0, 1}n → {0, 1}m of a given input x. We advocate the study of the complexity of generating the distribution h(x) for uniform x, given random bits. Our main results are: • There are explicit AC circuits of size poly(n) and depth O(1) whose output distribution has statistical distance 1/2n from the distribution (X, ∑ iXi) ∈ {0, 1}n × {0, 1, . . . , n} for uniform X ∈ {0, 1}n, despite the inability of these circuits to compute ∑ i xi given x. Previous examples of this phenomenon apply to different distributions such as (X, ∑ iXi mod 2) ∈ {0, 1}n+1. We also prove a lower bound independent from n on the statistical distance between the output distribution of NC circuits and the distribution (X,majority(X)). We show that 1− o(1) lower bounds for related distributions yield lower bounds for succinct data structures. • Uniform randomized AC circuits of poly(n) size and depth d = O(1) with error can be simulated by uniform randomized circuits of poly(n) size and depth d+ 1 with error + o(1) using ≤ (log n)O(log log n) random bits. Previous derandomizations [Ajtai and Wigderson ’85; Nisan ’91] increase the depth by a constant factor, or else have poor seed length. Given the right tools, the above results have technically simple proofs. ∗Supported by NSF grant CCF-0845003. Email: [email protected]