# Bounded Independence, AC0, and Random 3SAT

As reported here, here and here, Mark Braverman has just announced a proof of a 1990 conjecture by Linial and Nisan.

Mark proves that if ${C}$ is an AC0 boolean circuit (with NOT gates and with AND gates and OR gates of unbounded fan-in) of depth ${d}$ and size ${S}$, and if ${X}$ is any ${k}$-wise independent distribution with ${k = (\log S)^{O(d^2)}}$, then

$\displaystyle {\mathbb E} C(U_n) \approx {\mathbb E} C(X)$

that is, ${X}$ “fools” the circuit ${C}$ into thinking that ${X}$ is the uniform distribution ${U_n}$ over ${\{0,1\}^n}$. Plausibly, this might be true even for ${k = O((\log S)^{d-1})}$.

Nothing was known for depth 3 or more, and the depth-2 case was settled only recently by Bazzi, with a proof that, as you may remember, has been significantly simplified by Razborov about six months ago.

Mark’s proof relies on approximating ${C}$ via low-degree polynomials. The point is that if ${p}$ is an ${n}$-variate (real valued) polynomial of degree ${\leq k}$, and ${X}$ is a ${k}$-wise independent distribution ranging over ${\{0,1\}^n}$, then

$\displaystyle {\mathbb E} p(X) = {\mathbb E} p(U_n)$

Now if we could show that ${p}$ approximates ${C}$ both under ${X}$ and under ${U_n}$, in the sense that ${{\mathbb E} p(X) \approx {\mathbb E} C(X)}$, and also ${{\mathbb E} p(U_n) \approx {\mathbb E} C(U_n)}$, then we would be done.

The Razborov-Smolenski lower bound technique gives a probabilistic construction of a polynomial ${p}$ such that for every input ${x}$ one has a high probability that ${p(x) = C(x)}$. In particular, one get one polynomial ${p}$ such that both

$\displaystyle {\mathbb P}_{x\sim X} [p(x) = C(x) ] = 1-o(1) \ \ \ \ \ (1)$

and

$\displaystyle {\mathbb P}_{x\sim U_n} [p(x) = C(x) ] = 1-o(1) \ \ \ \ \ (2)$

Unfortunately this is not sufficient, because the polynomial ${p}$ might be very large at a few points, and so even if ${p}$ agrees with ${C}$ with high probability there is no guarantee that the average of ${p}$ is close to the average of ${C}$.

Using a result of Linial, Mansour and Nisan (developed in the context of learning theory), one can construct a different kind of low-degree approximating polynomial ${p}$, which is such that

$\displaystyle {\mathbb E}_{x\sim U_n} | C(x) - p(x)|^2 = o(1) \ \ \ \ \ (3)$

The Linial-Mansour-Nisan approximation, however, says nothing about the relation between ${p}$ and ${C}$ under the distribution ${X}$.

Using ideas of Bazzi’s, however, if we had a single polynomial ${p}$ such that properties (1), (2) and (3) are satisfied simultaneously, then we could construct another low-degree polynomial ${p'}$ such that ${{\mathbb E} p'(X) \approx {\mathbb E} C(X)}$, and also ${{\mathbb E} p'(U_n) \approx {\mathbb E} C(U_n)}$, giving us that ${C}$ is fooled by ${X}$.

As far as I understand, Mark constructs a polynomial satisfying properties (1), (2) and (3) by starting from the Razborov-Smolenski polynomial ${p}$, and then observing that the indicator function ${E}$ of the points on which ${p \neq C}$ is itself a boolean function admitting a Linial-Mansour-Nisan approximation ${e}$. Defining ${p' := p\cdot (1-e)}$, we have that ${p'}$ has all the required properties, because multiplying by ${1-e}$ “zeroes out” the points on which ${p}$ is excessively large.

I have been interested in this problem for some time because of a connection with the complexity of 3SAT on random instances.