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research
Quantified Derandomization of Linear Threshold Circuits
Authors
Bar-Yossef Z.
Bounded
+7Β more
Cheng Kuan
Impagliazzo R.
P
Pseudorandomness
Tamaki Suguru
The
Williams Ryan
Publication date
6 November 2017
Publisher
Doi
Cite
View
on
arXiv
Abstract
One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for
T
C
0
TC^0
T
C
0
, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for
T
C
0
TC^0
T
C
0
. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of
T
C
0
TC^0
T
C
0
circuits of depth
d
>
2
d>2
d
>
2
. Our first main result is a quantified derandomization algorithm for
T
C
0
TC^0
T
C
0
circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a
T
C
0
TC^0
T
C
0
circuit
C
C
C
over
n
n
n
input bits with depth
d
d
d
and
n
1
+
exp
β‘
(
β
d
)
n^{1+\exp(-d)}
n
1
+
e
x
p
(
β
d
)
wires, runs in almost-polynomial-time, and distinguishes between the case that
C
C
C
rejects at most
2
n
1
β
1
/
5
d
2^{n^{1-1/5d}}
2
n
1
β
1/5
d
inputs and the case that
C
C
C
accepts at most
2
n
1
β
1
/
5
d
2^{n^{1-1/5d}}
2
n
1
β
1/5
d
inputs. In fact, our algorithm works even when the circuit
C
C
C
is a linear threshold circuit, rather than just a
T
C
0
TC^0
T
C
0
circuit (i.e.,
C
C
C
is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of
T
C
0
TC^0
T
C
0
, and would consequently imply that
N
E
X
P
βΜΈ
T
C
0
NEXP\not\subseteq TC^0
NEXP
ξ
β
T
C
0
. Specifically, if there exists a quantified derandomization algorithm that gets as input a
T
C
0
TC^0
T
C
0
circuit with depth
d
d
d
and
n
1
+
O
(
1
/
d
)
n^{1+O(1/d)}
n
1
+
O
(
1/
d
)
wires (rather than
n
1
+
exp
β‘
(
β
d
)
n^{1+\exp(-d)}
n
1
+
e
x
p
(
β
d
)
wires), runs in time at most
2
n
exp
β‘
(
β
d
)
2^{n^{\exp(-d)}}
2
n
e
x
p
(
β
d
)
, and distinguishes between the case that
C
C
C
rejects at most
2
n
1
β
1
/
5
d
2^{n^{1-1/5d}}
2
n
1
β
1/5
d
inputs and the case that
C
C
C
accepts at most
2
n
1
β
1
/
5
d
2^{n^{1-1/5d}}
2
n
1
β
1/5
d
inputs, then there exists an algorithm with running time
2
n
1
β
Ξ©
(
1
)
2^{n^{1-\Omega(1)}}
2
n
1
β
Ξ©
(
1
)
for standard derandomization of
T
C
0
TC^0
T
C
0
.Comment: Changes in this revision: An additional result (a PRG for quantified derandomization of depth-2 LTF circuits); rewrite of some of the exposition; minor correction
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Last time updated on 10/08/2021