The complexity of DNF of parities

We study depth 3 circuits of the form OR AND XOR, or equivalently-DNF of parities. This model was first explicitly studied by Jukna (CPC'06) who obtained a 2(n) lower bound for explicit functions in this model. Several related models have gained attention in the last few years, such as parity decision trees, the parity kill number and AC0 XOR circuits. For a function f : f0; 1gn ! f0; 1g, we denote by DNFδ(f) the least integer s for which there exists an OR AND XOR circuit, with OR gate of fan-in s, that computes f. We summarize some of our results: For any affine disperser f : f0; 1gn ! f0; 1g for dimension k, it holds that DNF (f) ≥ 2^n-2k. By plugging Shaltiel's affine disperser (FOCS'11) we obtain an explicit 2^n-no(1) lower bound. We give a non-Trivial general upper bound by showing that DNF (f) ≥ O(2n=n) for any function f on n bits. This bound is shown to be tight up to an O(log n) factor. We show that for any symmetric function f it holds that DNF(f) ≤ 1:5ⁿ poly(n). Furthermore, there exists a symmetric function f for which this bound is tight up to a polynomial factor. We show tighter bounds for symmetric threshold functions. For example, we show that the majority function has DNF complexity of ω2n=2 poly(n). This is also tight up to a polynomial factor. For the inner product function IP on n inputs we show that DNFδ(IP) = 2ⁿ⁼²-1. Previously, Jukna gave a lower bound of (2n=4) for the DNF complexity of this function. We further give bounds for any low degree polynomial. Finally, we obtain a 2^n-o(n) average case lower bound for the parity decision tree model using affine extractors.