Affiliation:
1. Northeastern University, Boston, MA
Abstract
We prove that for every distribution
D
on
n
bits with Shannon entropy ≥
n
−
a
, at most
O
(2
d
a
log
d
+1
g
)/γ
5
of the bits
D
i
can be predicted with advantage γ by an AC
0
circuit of size
g
and depth
D
that is a function of all of the bits of
D
except
D
i
. This answers a question by Meir and Wigderson, who proved a corresponding result for decision trees.
We also show that there are distributions
D
with entropy ≥
n
−
O
(1) such that any subset of
O
(
n
/ log
n
) bits of
D
on can be distinguished from uniform by a circuit of depth 2 and size poly(
n
). This separates the notions of predictability and distinguishability in this context.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
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