Poly Logarithmic Independence Fools Ac Circuits

Poly-logarithmic independence fools AC0 circuits
Mark Braverman
Microsoft Research New England
January 30, 2009
Abstract
We prove that poly-sized AC0 circuits cannot distinguish a poly-logarithmically in-
dependent distribution from the uniform one. This settles the 1990 conjecture by
Linial and Nisan [LN90].
The only prior progress on the problem was by Bazzi
[Baz07, Baz09], who showed that O(log2n)-independent distributions fool poly-size
DNF formulas. Razborov [Raz08] has later given a much simpler proof for Bazzi’s
theorem.
1
Introduction
1.1
The problem
The main problem we consider is on the power of r-independence to fool AC0 circuits. For
a distribution µ on the ﬁnite support {0, 1}n, we denote by Eµ[F ] the expected value of F
on inputs drawn according to µ. For an event X, we denote by µ[X] its probability under µ.
When the distribution under consideration is the uniform distribution on {0, 1}n, we suppress
the subscript and let E[F ] denote the expected value of F , and P[X] the probability of X.
A distribution µ is said to ε-fool a function F if
|Eµ[F ] − E[F ]| < ε.
The distribution µ on {0, 1}n is r-independent if every restriction of µ to r coordinates
is uniform on {0, 1}r. AC0 circuits are circuits with AN D, OR and N OT gates, where
the fan-in of the gates is unbounded. The depth of a circuit C is the maximum number of
AN D/OR gates between an input of C and its output. The problem we study is
Main Problem. How large does r = r(m, d, ε) have to be in order for every r-independent
distribution µ on {0, 1}n to ε-fool every function F that is computed by a depth-d AC0 circuit
of size ≤ m?
1

The original conjecture by [LN90] was that for a constant ε, r(m, d, ε) = (log m)d−1
suﬃces. The conjecture with these parameters turned out to be false [LV96], but it remained
open whether r(m, d, ε) = (log m)O(1) for a constant d and ε.
Prior to our work, Bazzi [Baz07, Baz09], in a proof that was later simpliﬁed by Razborov
[Raz08], showed that a poly-logarithmic r is suﬃcient for d = 2 (i.e. when the F ’s are DNF
or CNF formulas):
Theorem 1. [Baz07, Raz08] r(m, 2, ε)-independence ε-fools depth-2 circuits, where
m
r(m, 2, ε) = O log2
.
ε
Our main result is that for any constant d, r(m, d, ε) is poly-logarithmic in m/ε. This
gives a huge class of distributions that look random to AC0 circuits. For example, as in
[Baz09], it implies that linear codes with poly-logarithmic seed length can be PRGs for AC0.
1.2
Main results
We prove the following:
Main Theorem. Let s ≥ log m be any parameter. Let F be a boolean function computed by
a circuit of depth d and size m. Let µ be an r-independent distribution where
r ≥ r(s, d) = 3 · 60d+3 · (log m)(d+1)(d+3) · sd(d+3),
then
|Eµ[F ] − E[F ]| < ε(s, d),
where ε(s, d) = 0.82s · (15m).
In particular, by taking s = 5 log 15m , we get the following:
ε
Corollary 2. r(m, d, ε)-independence ε-fools depth-d AC0 circuits of size m, where
15m d(d+3)
m O(d2)
r(m, d, ε) = 3 · 60d+3 · (log m)(d+1)(d+3) ·
5 log
= log
.
ε
ε
Note that by choosing ε = 2−nδ for a small δ = δ(d), one sees that polynomial indepen-
dence fools AC0 circuits up to an exponentially small error. The results carry some meaning
√
for super-constant d’s up to d = ˜
O( log m).
As in [Baz09], we can use [AGM02] to show that almost r-independent distributions also
fool AC0. A distribution µ is called a (δ, r)-approximation, if µ is δ-close to uniform for
every r (distinct) coordinates. Thus an r-independent distribution is a (0, r)-approximation.
We use the following theorem.
Theorem 3. [AGM02] Let µ be a (δ, r)-approximation over n variables. Then µ is nr · δ-
close to an r-independent distribution µ .
2

Theorem 3 and the Main Theorem immediately imply:
Corollary 4. For every boolean circuit F of depth d and size m over n variables and any
s ≥ log m, let
r ≥ r(s, d) = 3 · 60d+3 · (log m)(d+1)(d+3) · sd(d+3).
Then for any (δ, r)-approximation µ,
|Eµ[F ] − E[F ]| < ε(s, d) + nr · δ = 0.82s · (15m) + nr · δ.
Corollary 4 in turn implies:
Corollary 5. (δ, r(m, d, ε))-approximations ε-fool depth-d AC0 circuits of size m, where
15m d(d+3)
r(m, d, ε) = 4 · 60d+3 · (log m)(d+1)(d+3) ·
5 log
,
ε
as long as δ is suﬃciently small so that
ε > 2nr(m,d,ε).
δ
1.3
Techniques and proof outline
As in [Baz09], our strategy is to approximate F with low degree polynomials over R. The
reason being that degree-r polynomials are completely fooled by r-independence.
Proposition 6. Let f :
n
R
→ R be a degree-r polynomial, and let µ be an r-independent
distribution. Then f is completely fooled by µ:
Eµ[f ] = E[f ].
Proposition 6 is true by linearity of expectation, since every term of f is a product of
≤ r variables, whose distribution is uniform under µ.
In our construction we combine two types of approximations of AC0 circuits by low degree
polynomials over R. The ﬁrst one is combinatorial in the spirit of [Raz87, Smo87, BRS91,
Tar93] (for a comprehensive survey on polynomials in circuit complexity see e.g. [Bei93]).
These approximating polynomials agree with F on all but a small fraction of inputs. Thus
for such a polynomial f , P[f = F ] is very close to 1. While essentially using the same
construction as [BRS91, Tar93], utilizing tools from [VV85], we repeat the construction
from scratch in Lemma 8, since we want to reason about details of the construction. We
believe that any construction in this spirit would ﬁt in our proof.
The second approximation is based on Fourier analysis and uses [LMN93] that shows
that any AC0 function G can be approximated by a low degree polynomial g so that the 2
norm G − g 2 is small. There is no guarantee, however, that g agrees with G on any input
2
(most likely, it doesn’t), or that G − g 1 is small.
3

We use an approximation f of F of the ﬁrst type as the starting point of our construction.
Thus P[f = F ] is very small. If we knew that F −f 2 is small we would be done by a simple
2
argument similar to one that appeared in [Baz09]. Unfortunately, there are no guarantees,
that f is close to F on average, since f may deviate wildly on points where f = F (in fact,
it is likely untrue that F − f 2 is small).
2
Our key insight is that in the construction of f , the indicator function E of where f fails
to agree with F is an AC0 function itself. Thus E = 1 whenever f = F , and P[E = 1] is
very small (since f = F most of the time). We then use a low-degree approximation ˜
E of E
of the second type so that
˜
E − E 2 is very small. We then take f = f · (1 − ˜
E). The idea
2
is that 1 − ˜
E ≈ 1 − E will kill the values of f where it misbehaves (and thus E = 1), while
leaving other values (where E = 0) almost unchanged. Note that the values where f = 0
remain completely unchanged, and thus f is a semi-exact approximation of F . In Lemma
10 we show that F − f 2 is small. We choose f to “almost agree” with F against both
2
the uniform distribution and the distribution µ, a property we use to ﬁnish the proof.
It should be noted that while an inductive proof on the depth d of F is a natural approach
to the problem, a non-inductive construction appears to yield much better parameters for
the theorem.
1.4
Paper organization
The rest of the paper is organized as follows. In Section 2 we repeat the [LMN93] theorem
on low-degree 2-approximation, and develop low-degree approximation tools that are used
in the proof of the main theorem. In Section 3 we prove the main theorem.
Acknowledgments
I am grateful to Louay Bazzi, Marek Karpinski, and Alex Samorodnitsky for stimulating
discussions at the early stages of this work. I would like to thank Nina Balcan, Henry Cohn,
Nick Harvey, Toni Pitassi, Madhu Sudan, and many others for the many useful comments
on the earlier versions of this manuscript.
2
Semi-exact approximations and error functions
We will make use of the [LMN93] bound:
Lemma 7. ([LMN93]) If F : {0, 1}n → {0, 1} is a boolean function computable by a depth-d
circuit of size m, then for every t there is a degree t polynomial ˜
f with
1
F − ˜
f 2 =
|F (x) − ˜
f (x)|2 ≤ 2m · 2−t1/d/20.
2
2n x∈{0,1}n
As a ﬁrst step, we prove the following lemma:
4

Lemma 8. Let ν be any probability distribution on {0, 1}n. For a circuit of depth d and size
m computing a function F , for any s, there is a degree r = (s · log m)d polynomial f and a
depth < d + 3 boolean function Eν of size O(m2r) such that
• ν[Eν(x) = 1] < 0.82sm, and
• whenever Eν(x) = 0, f (x) = F (x).
Thus Eν tells us whether there is a mistake in f , and the weight of the mistakes as
measured by ν is very small. Note that when there is a mistake, f does not have to be equal
to 1 − F , and can actually be quite large in absolute value.
Proof. We construct the polynomial f by induction on d, and show that w.h.p. f = F . The
function Eν follows from the construction.
We will show how to make a step with an AN D gate. Since the whole construction is
symmetric with respect to 0 and 1, the step also holds for an OR gate. Let
F = G1 ∧ G2 ∧ . . . ∧ Gk,
where k < m. For convenience, let us assume that k = 2 is a power of 2. We take a
collection of
t := s log m
random Poisson subsets of {1, 2, . . . , k}: at least s of each of the p = 2−1, 2−2, . . . , 2− = 1/k.
Denote the sets by S1, . . . , St – we ignore empty sets. In addition, we make sure to include
{1, . . . , k} as one of the sets. Let g1, . . . , gk be the approximating polynomials for G1, . . . , Gk.
We set
t
f :=
gj − |Si| + 1 .
i=1
j∈Si
By the induction assumption, the degrees of gj are d ≤ (s · log m)d−1, hence the degree of
f is bounded by t · d ≤ (s · log m)d. Next we bound the error P[f = F ]. It consists of two
terms:
t
k
ν[f = F ] ≤ ν[gj = Gj for some j] + ν
Gj − |Si| + 1
=
Gj .
(1)
i=1
j∈Si
j=1
In other words, to make a mistake, either one of the inputs has to be dirty, or the approxi-
mating function for the AND has to make a mistake. We will focus on the second term. The
ﬁrst term is bounded by union bound. We ﬁx a vector of speciﬁc values G1(x), . . . , Gk(x),
and calculate the probability of an error over the possible choices of the random sets Si.
Note that if all the Gj(x)’s are 1 then the value of F (x) = 1 is calculated correctly
with probability 1. Suppose that F (x) = 0 (and thus at least one of the Gj(x)’s is 0).
Let 1 ≤ z ≤ k be the number of zeros among G1(x), . . . , Gk(x). And α be such that
2α ≤ z < 2α+1. Let S be a random Poisson set with p = 2−α−1. Our formula will work
5

correctly if S hits exactly one 0 among the z zeros of G1(x), . . . , Gk(x). The probability of
this event is exactly
1
1
z · p · (1 − p)z−1 ≥
· (1 − p)1/p−1 >
> 0.18.
2
2e
Hence the probability of being wrong after s such sets is bounded by 0.82s. Since this is true
for any value of x, we can ﬁnd a collection of sets Si such that the probability of error is at
most 0.82s according to ν. By making the same probabilistic argument at every node, we
get an error of < 0.82sm by union bound.
Finally, if we know the sets Si at every node, it is easy to check whether there is a mistake
by checking that no set contains exactly one 0, thus yielding the depth < (d + 3) function
Eν.
In addition, we have:
Proposition 9. In Lemma 8, for s ≥ log m, f ∞ < (2m)deg(f)−2 = (2m)td−2.
Proof. We prove the statement by induction on d. For d = 1, deg(f ) = t and the functions
gj are just 0/1-valued literals. Since |Si| ≤ m for all i, we have for every x:
t
|f (x)| =
gj − |Si| + 1 ≤ mt < (2m)t−2.
i=1 j∈Si
For the step, assuming the statement is true for d − 1 ≥ 1, we have
t
f ∞ ≤
gj ∞ + |Si| − 1 <
i=1 j∈Si
t
t
m · (2m)td−1−2 + m < (2m)td−1−1
< (2m)td−2.
i=1
Applying results from [LMN93] we can now take any shallow function F and modify it a
little bit, so that the modiﬁed function would have a good one-sided-error approximation:
Lemma 10. Let F be computed by a circuit of depth d and size m.
Let s1, s2 be two
parameters with s1 ≥ log m. Let µ be any probability distribution on {0, 1}n. Set
1
ν :=
(µ + U{0,1}n).
2
Let Eν be the function from Lemma 8 with s = s1. Set F = F ∨ Eν. Then there is a
polynomial f of degree rf ≤ (s1 · log m)d + s2, such that
• P[F = F ] < 2 · 0.82s1m;
6

• µ[F = F ] < 2 · 0.82s1m;
1/(d+3)
•
F − f 2 < 0.82s1 · (4m) + 22.9(s1·log m)d log m−s
/20
2
, and
2
• f (x) = 0 whenever F (x) = 0.
Proof. The ﬁrst two properties follow from Lemma 8 directly, since
P[Eν = 1], µ[Eν = 1] ≤ 2 · ν[Eν = 1] < 2 · 0.82s1m.
Let f be the approximating polynomial for F from that lemma, so that F = F = f
whenever Eν = 0, and thus f = 0 whenever F = 0. By Proposition 9 we have
f ∞ < (2m)(s1·log m)d < 21.4(s1·log m)d log m.
We let ˜
Eν be the low degree approximation of Eν of degree s2. By [LMN93] (Lemma 7), we
have
1/(d+3)
E
2
/20
2
ν − ˜
Eν
< O(m3) · 2−s
.
2
Let
f := f · (1 − ˜
Eν).
Then f = 0 whenever F = 0. It remains to estimate F − f 2:
2
F − f 2 ≤ 2 · F − f · (1 − E
+ 2 · f · (1 − E
= 2 · E
2 + 2 · f · (E
≤
2
ν ) 2
2
ν ) − f
2
2
ν 2
ν − ˜
Eν) 22
1/(d+3)
2 · P[E
2
/20
2
ν = 1] + 2 ·
f 2 · E
∞
ν − ˜
Eν
< 0.82s1(4m) + 22.9(s1·log m)d log m−s
,
2
which completes the proof.
3
Main Theorem
Lemma 10 implies the following:
Lemma 11. For every boolean circuit F of depth d and size m and any s ≥ log m, and for
any probability distribution µ on {0, 1} there is a boolean function F and a polynomial f of
l
degree less than
rf = 3 · 60d+3 · (log m)(d+1)(d+3) · sd(d+3)
such that
• µ[F = F ] < ε(s, d)/3,
• P[F = F ] < ε(s, d)/3,
• f ≤ F on {0, 1}n, and
l
• E[F − f ] < ε(s, d)/3,
l
7

where ε(s, d) = 0.82s · (15m).
Proof. Let F be the boolean function and let f be the polynomial from Lemma 10 with
s1 = s and s2 ≈ 60d+3 · (log m)(d+1)(d+3) · sd(d+3). The ﬁrst two properties follow directly from
the lemma. Set
f := 1 − (1 − f )2.
l
It is clear that f ≤ 1 and moreover f = 0 whenever F = 0, hence f ≤ F . Finally,
l
l
l
F (x) − f (x) = 0 when F (x) = 0, and is equal to
l
F (x) − f (x) = (1 − f (x))2 = (F (x) − f (x))2
l
when F (x) = 1, thus
E[F − f ] ≤ F − f 2 < 0.82s · (5m) = ε(s, d)/3,
l
2
by Lemma 10. To ﬁnish the proof we note that the degree of f is bounded by
l
2 · ((s1 · log m)d + s2) < 2.5 · s2 < rf .
Lemma 11 implies the following:
Lemma 12. Let s ≥ log m be any parameter. Let F be a boolean function computed by a
circuit of depth d and size m. Let µ be an r-independent distribution where
r ≥ 3 · 60d+3 · (log m)(d+1)(d+3) · sd(d+3),
then
Eµ[F ] > E[F ] − ε(s, d),
where ε(s, d) = 0.82s · (15m).
Proof. Let F be the boolean function and let f be the polynomial from Lemma 11. The
l
degree of f is < r. We use the fact that since µ is r-independent, Eµ[f ] = E[f ] (see
l
l
Proposition 6 above):
Eµ[F ] ≥ Eµ[F ] − µ[F = F ] ≥ Eµ[f ] − ε(s, d)/3 = E[f ] − ε(s, d)/3 =
l
l
E[F ] − E[F − f ] − ε(s, d)/3 > E[F ] − 2ε(s, d)/3 ≥
l
E[F ] − P[F = F ] − 2ε(s, d)/3 > E[F ] − ε(s, d).
The dual inequality to Lemma 12 follows immediately by applying the lemma to the
negation F = 1 − F of F . We have Eµ[F ] > E[F ] − ε(s, d), and thus
Eµ[F ] = 1 − Eµ[F ] < 1 − E[F ] + ε(s, d) = E[F ] + ε(s, d).
Together, these two statements yield the main theorem:
8

Main Theorem. Let s ≥ log m be any parameter. Let F be a boolean function computed by
a circuit of depth d and size m. Let µ be an r-independent distribution where
r ≥ r(s, d) = 3 · 60d+3 · (log m)(d+1)(d+3) · sd(d+3),
then
|Eµ[F ] − E[F ]| < ε(s, d),
where ε(s, d) = 0.82s · (15m).
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10