Abstract
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 pseudorandom generators ( PRG s )) algorithms for FORMULA[n 1.99 ]∘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 the following: • The Generalized Inner Product function GIP k n cannot be computed in FORMULA[s]° G on more than 1/2+ε fraction of inputs for s=o(n 2 /k⋅4 k ⋅R (k) (G)⋅log(n/ε)⋅log(1/ε)) 2 ). This significantly extends the lower bounds against bipartite formulas obtained by [62]. As a corollary, we get an average-case lower bound for GIP k n against FORMULA[n 1.99 ]∘PTF k −1 , i.e., sub-quadratic-size De Morgan formulas with degree-k-1) PTF ( polynomial threshold function ) gates at the bottom. Previously, it was open whether a super-linear lower bound holds for AND of PTFs. • 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 the special case of FORMULA[s]∘LTF, i.e., size- s formulas with LTF ( linear threshold function ) gates at the bottom, we get the better seed length O(n 1/2 ⋅s 1/4 ⋅log(n)⋅log(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n halfspaces in the regime where ε≤1/n, complementing a recent result of [45]. • There exists a randomized 2 n-t #SAT algorithm for FORMULA[s]∘G, where t=Ω(n\√s⋅log 2 (s)⋅R (2) (G))/1/2. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n 1.99 ]∘LTF. • The Minimum Circuit Size Problem is not in FORMULA[n 1.99 ]∘XOR; thereby making progress on hardness magnification, in connection with results from [14, 46]. On the algorithmic side, we show that the concept class FORMULA[n 1.99 ]∘XOR can be PAC-learned in time 2 O(n/log n) .
Highlights
A Boolean formula over {0, 1}-valued input variables x1, . . . , xn is a binary tree whose internal nodes are labelled by AND or OR gates, and whose leaves are marked with a variable or its negation
We let FORMULA[s] ◦ G denote the set of Boolean functions computed by formulas containing at most s leaves, where each leaf computes according to some function in G
For a Boolean function g : {0, 1}n → {0, 1}, we denote by Rδ(k)(g) the communication cost of the best k-party number-on-forehead (NOF) communication protocol that computes g with probability at least 1 − δ on every input, where the probability is taken over the random choices of the protocol
Summary
A (de Morgan) Boolean formula over {0, 1}-valued input variables x1, . . . , xn is a binary tree whose internal nodes are labelled by AND or OR gates, and whose leaves are marked with a variable or its negation. Motivated by several recent works, we initiate a systematic study of the FORMULA ◦ G model, i.e., Boolean formulas whose leaves are labelled by an arbitrary function from a fixed class G. This model unifies and generalizes a variety of models that have been previously studied in the literature: Oliveira, Pich, and Santhanam [45] show that proving certain lower bounds against formulas of size n1+ε over parity (XOR) gates would have significant consequences in complexity theory. Abboud and Bringmann [1] consider formulas where the leaves are threshold gates whose input wires can be arbitrary functions applied to either the first or the second half of the input. We show that this perspective leads to stronger lower bounds, general satisfiability algorithms, and better pseudorandom generators for a broad class of functions
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