The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by
deterministic
algorithms, if one can obtain
impressive
(i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form
n
1+
ϵ
would suffice to derandomize interesting classes of probabilistic algorithms. We show the following:
— If the word problem over
S
5
requires constant-depth threshold circuits of size
n
1+
ϵ
for some
ϵ
>0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size).
— If there are no constant-depth arithmetic circuits of size
n
1+
ϵ
for the problem of multiplying a sequence of
n
3×3 matrices, then, for every constant
d
, black-box identity testing for depth-
d
arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC
0
circuits of subexponential size).
Lower Bounds for Arithmetic Circuits via the Hankel Matrix
