Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

The class FORMULA[s]∘G consists of Boolean functions computable by size- s De Morgan formulas whose leaves are any from a G. We give lower bounds and (SAT, Learning, pseudorandom generators ( PRG )) algorithms for FORMULA[n 1.99 ]∘G, classes G with low communication complexity . Let R (k) be the maximum k -party number-on-forehead randomized function in Among other results, we show following: • Generalized Inner Product GIP n cannot computed FORMULA[s]° on more than 1/2+ε fraction inputs s=o(n 2 /k⋅4 ⋅R (G)⋅log⁡(n/ε)⋅log⁡(1/ε)) ). This significantly extends against bipartite obtained [62]. As corollary, get an average-case bound ]∘PTF −1 , i.e., sub-quadratic-size degree-k-1) PTF polynomial threshold ) gates at bottom. Previously, it was open whether super-linear holds AND PTFs. There is seed length n/2+O(s⋅R (2) (G)⋅log⁡(s/ε)⋅log⁡(1/ε)) that ε-fools FORMULA[s]∘G. For special case FORMULA[s]∘LTF, LTF linear bottom, better O(n 1/2 ⋅s 1/4 ⋅log⁡(n)⋅log⁡(n/ε)). In particular, this provides first non-trivial (with o(n)) intersections halfspaces regime where ε≤1/n, complementing recent result [45]. exists n-t #SAT algorithm FORMULA[s]∘G, t=Ω(n\√s⋅log ⁡(s)⋅R (G))/1/2. implies nontrivial ]∘LTF. Minimum Circuit Size Problem not ]∘XOR; thereby making progress hardness magnification, connection results [14, 46]. On algorithmic side, concept ]∘XOR can PAC-learned time O(n/log n)