Quantified Derandomization of Linear Threshold Circuits · arXivDesk
1709.07635Sep 22, 2017Changes in this revision: An additional result (a PRG for quantified derandomization of depth-2 LTF circuits); rewrite of some of the exposition; minor corrections
Quantified Derandomization of Linear Threshold Circuits
One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for TC0, 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 TC0. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of TC0
Nearby in the stack
circuits of depth
d>2
. Our first main result is a quantified derandomization algorithm for
TC0
circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a
TC0
circuit
C
over
n
input bits with depth
d
and
n1+exp(−d)
wires, runs in almost-polynomial-time, and distinguishes between the case that
C
rejects at most
2n1−1/5d
inputs and the case that
C
accepts at most
2n1−1/5d
inputs. In fact, our algorithm works even when the circuit
C
is a linear threshold circuit, rather than just a
TC0
circuit (i.e.,
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
TC0
, and would consequently imply that
NEXP⊆TC0
. Specifically, if there exists a quantified derandomization algorithm that gets as input a
TC0
circuit with depth
d
and
n1+O(1/d)
wires (rather than
n1+exp(−d)
wires), runs in time at most
2nexp(−d)
, and distinguishes between the case that
C
rejects at most
2n1−1/5d
inputs and the case that
C
accepts at most
2n1−1/5d
inputs, then there exists an algorithm with running time