Algorithms and lower bounds for de morgan formulas of low-communication leaf gates

The class FORMULA[s] ο G consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class G. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n1.99] ο G, for classes G of functions with low communication complexity. Let R(k) (G) be the maximum k-party number-on-forehead randomized communication complexity of a function in G. Among other results, we show that: • The Generalized Inner Product function GIPkn cannot be computed in FORMULA[s] ο G on more than 1/2 + ε fraction of inputs for [MATH HERE] This significantly extends the lower bounds against bipartite formulas obtained by [61]. As a corollary, we get an average-case lower bound for GIPkn against FORMULA[n1.99] ο PTFk-1, i.e., sub-quadratic-size de Morgan formulas with degree-(k - 1) PTF (polynomial threshold function) gates at the bottom. • There is a PRG of seed length [MATH HERE] that ε-fools FORMULA[s] ο G. For the special case of FORMULA[s] οLTF, i.e., size-s formulas with LTF (linear threshold function) gates at the bottom, we get the better seed length [MATH HERE]. In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n half-spaces in the regime where ε ≤ 1/n, complementing a recent result of [44]. • There exists a randomized 2n-t-time #SAT algorithm for FORMULA[s] ο G, where [MATH HERE] In particular, this implies a nontrivial #SAT algorithm for FORMULA[n1.99] ο LTF. • The Minimum Circuit Size Problem is not in FORMULA[n1.99] ο XOR; thereby making progress on hardness magnification, in connection with results from [45, 12]. On the algorithmic side, we show that the concept class FORMULA[n1.99] ο XOR can be PAC-learned in time 2o(n/log n).