Affiliation:
1. Simon Fraser University, Burnaby, BC, Canada
2. University of Warwick, Coventry, UK
3. Imperial College London, London, UK
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)
.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Cited by
2 articles.
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