The class FORMULA[s]∘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]∘G, for classes
G of functions with low communication complexity. Let
R(k)(G) be the maximum k-party NOF randomized communication
complexity of G. We show:
(1) The Generalized Inner Product function GIPnk cannot be computed in
FORMULA[s]∘G on more than 1/2+ε fraction of inputs
for s=o((k⋅4k⋅R(k)(G)⋅log(n/ε)⋅log(1/ε))2n2). As a corollary, we get an average-case lower bound for
GIPnk against FORMULA[n1.99]∘PTFk−1.
(2) There is a PRG of seed length n/2+O(s⋅R(2)(G)⋅log(s/ε)⋅log(1/ε)) that ε-fools FORMULA[s]∘G. For
FORMULA[s]∘LTF, we get the better seed length O(n1/2⋅s1/4⋅log(n)⋅log(n/ε)). This gives the first
non-trivial PRG (with seed length o(n)) for intersections of n half-spaces
in the regime where ε≤1/n.
(3) There is a randomized 2n−t-time #SAT algorithm for FORMULA[s]∘G, where t=Ω(s⋅log2(s)⋅R(2)(G)n)1/2. In particular, this implies a nontrivial
#SAT algorithm for FORMULA[n1.99]∘LTF.
(4) The Minimum Circuit Size Problem is not in FORMULA[n1.99]∘XOR.
On the algorithmic side, we show that FORMULA[n1.99]∘XOR can be
PAC-learned in time 2O(n/logn)