Poly-logarithmic independence fools circuits, a survey

POLY-LOGARITHMIC

INDEPENDENCE FOOLS

CIRCUITS, A SURVEY

Mirmojtaba (Mojtaba) Gharibi

December 2010

OVERVIEW

In 1990, Linial and Nisan conjectured that no

circuit can distinguish the uniform

distribution from any poly-logarithmic

independent distribution

In a recent breakthrough, after about two

decades of no progress, the conjecture was

settled by Mark Braverman

OVERVIEW : CIRCUITS

A circuit consisting of polynomial number of ,

and gates.

gates only appear at the input nodes

Depth of the circuit is constant and is the

number of gates from the input to the output

OVERVIEW:R-INDEPENDENT DISTRIBUTION

Consider a set of random variables distributed

according to an r-independent distribution over .

Looking at any subset of size at most , the

probability that any bit in that subset be is

independent of any other outcomes in that subset

AN EXAMPLE

Consider the following probability distribution over :

where are set uniformly and independently at random.

This is an example of a -independent distribution, since choosing any subset of size at most , each variable is totally independent from any outcome within that set:

AN EXAMPLE

Subsets of size at most of are:

OVERVIEW:R-INDEPENDENT DISTRIBUTION

So, -independent distributions are in some

sense locally random

Whereas on the other hand, the uniform

distribution is globally random

LINIAL AND NISAN 1990’S CONJECTURE

circuit cannot distinguish local randomness

from global randomness.

i.e. any function computable by an circuit

aiming to distinguish the Uniform distribution

from an -independent distribution output the

same thing on inputs drawn from both

distributions except with a negligible bias.

A DEFINITION

A distribution is said to the Boolean function

if:

Or equivalently

LINIAL AND NISAN 1990’S CONJECTURE

We are interested to know how large needs

to be in order for to any circuit of size

operating on

LN conjecture (with relaxed parameters):

SMALL BIAS

So for a polynomially small , a polylogarithmic is good enough.

We cannot take for granted that a polynomially small , can be boosted to a constant probability by taking majority. Why? circuits are not capable of taking majority.

Polynomially small bias is taken as negligible for circuits.

MOTIVATION AND APPLICATIONS

It says that if an circuit accepts truly random bits,

then it also accepts pseudorandom bits. So if one

wants to distinguish random bits from

pseudorandom bits, he needs a more powerful

circuit, possibly with exponentially more gates or

more powerful gates like XOR and MAJORITY or with

greater depth (e.g. logarithmic).

So it gives a better understanding of limitations of an

important complexity class. The result may be later

used as a tool for proving lower bounds.

MOTIVATION AND APPLICATIONS

Since 1980’s we have known many serious

limitations of circuits like many specific

pseudorandom distributions that fool

circuits. The conjecture actually says that

any r-independent distribution will fool

them. So this gives a large class of

distributions that look random to circuits.

For instance, linear codes with poly-logarithmic

seed length can be PRGs for .

HISTORY

In 2007, in the first noticeable development,

Bazzi settled the conjecture for circuits. His proof

was about 50 pages.

In 2008, Razborov simplified Bazzi’s proof to a 3-

page proof.

Finally, in 2009, Mark Braverman settled the

conjecture with a short proof.

BAZZI’S RESULTS

Bazzi’s theorem: - independence depth 2

circuits where

BAZZI’S RESULTS

Bazzi’s proof was based on harmonic and poset

analysis techniques. He also used Linial,

Mansour and Nisan celebrated result of 1993 of

low-degree real polynomial approximation.

Razborov’s proof does not use Fourier analysis

techniques except of making connection to

Linial, Mansour, and Nisan’s theorem.

REST OF THIS PRESENTATION

Razborov-Smolensky’s approximation

technique by low-degree polynomials over

finite fields

Linial, Mansour and Nisan’s approximation

technique by low-degree real polynomials

Mark Braverman’s proof of the conjecture

RAZBOROV-SMOLENSKY’S APPROXIMATION

Recall : the technique was used in the class to prove

It involves approximating a Boolean function using a low-degree polynomial over finite fields.

Then, knowing the properties of the low-degree polynomials, we can talk about the properties of F.


For example, for PARITY:

Any function can be well approximated by low-

degree polynomials

can be represented with a high-degree

polynomial

A low-degree polynomial cannot approximate a

high-degree polynomial

Hence


Denote the approximation of Boolean

function with a low-degree polynomial .

In Rozborov-Smolenskey’s technique, the

criteria for a good approximation is that for

a large fraction of inputs. However,

when , they may largely disagree.


0

0

1

1

𝑭

𝒇

LINIAL, MANSOUR AND NISAN’S APPROXIMATION

Denote the approximator of the Boolean

function with a low-degree real polynomial

LMN says that approximation of is possible

via low-degree real polynomials. But there

is no guarantee that on any inputs.

Most likely for any inputs,


[LMN93]: Every Boolean function

computable by an circuit of size and depth ,

can be approximated by a real low-degree

polynomial of degree :


0

1𝑭

𝒇

THE CONJECTURE

We wish to prove : For any -independent distribution where :

THE BASIC IDEA

If we can find a “good” low-degree polynomial

approximation of F we are done.

Because:

Low-degree polynomials are composed of low-degree

terms.

The expectation of any polynomial is the sum of the

expectation of its terms.

Each term’s expectation is exactly the same under

and distribution. So the polynomial’s expectation is

also the same.

THE FIRST STEP IN THE PROOF

We will construct a distribution on the

polynomial over a proper finite field such

that with high probability agrees with on

any given input. So for any given measure ,

with high probability we have an

approximator having a small error, which

implies that there exists a specific

approximator having a small error with

respect to .


If the polynomial is a good approximator (i.e.

for some small ), one can transform to a

good approximator and show that the

conjecture holds.

However, most likely it is not the case! Why?

0

0

1

1

𝑭

𝒇

Small fraction


We want a good approximation too, but is

behaving wildly in its bad region.

Let’s first construct , then we will deal with

this problem in the second step of the proof!

APPROXIMATOR’S CONSTRUCTION

NOT gates will all appear at input nodes

which are easy to approximate.

Anywhere else, we have AND/OR gates.

As usual we use induction on the depth of

the circuit.

We describe the construction of AND

approximator. OR’s construction follows from

the symmetry between 1 & 0 and AND & OR.

AND’S APPROXIMATOR

Consider set of indexes . For a parameter , we prepare a collection of of its subsets in the following way:

For each of we prepare at least random subsets of

We include in each subset each of the indexes with independent probability . We also include . Denote these subsets with

⋀𝐺1 𝐺𝑘…

𝐹

For convenience let us assume is a power of 2, e.g. .


Construct:

is the approximator of .

Let us for now focus on the case that all have approximated correctly. We later bound all the errors by the union bound.


approximates correctly when .

However, we may err when .


When , let us say of the ’s have had been zero.( is

the number of zeros). approximates correctly if at

least one of hits exactly one zero.

The probability of a wrong approximation can be

shown to be at most . Since it is true for any value

of , we can actually find a collection of that yields

that error bound.

By the union bound

The degree of the polynomial is

INSIGHT

Our goal was to make our behave nicely in its bad region.

Here is the idea:

Given our choices for , there exist another

Boolean formula computable by an circuit of

slightly more depth and size which can

determine if has erred or not. Denote it with .

0

0

1

1

𝑭

𝒇

0

1 ℰ𝝂

INSIGHT

How can we use this to make have a better behaviour in its bad region?

Set Compute .

0

0

1

1

𝑭

𝒇

0

1 ℰ𝝂

0

1𝑭 ′

INSIGHT

One can show that is a good approximator of

with respect to both measures and .

and are small:

Also is a good approximator of . But it is

Boolean, not a polynomial. We wish was

exactly behaving like !!!

THE SECOND STEP IN THE PROOF

Here is our new strategy:

Since we failed to find a good approximator of

directly, we try to find a good low-degree

approximator of which we denote by .

Since is a good approximator of , is also a good

approximator of .

By a good approximator we mean an approximator

which by known techniques can be transformed into

an approximator.


We approximate with a low-degree real polynomial of degree based on Linial, Nisan and Mansour technique. Denote the approximation by .

We use this approximation to form .

We choose t large enough to have close to 0 when is close to 1.

0

1

𝟏− ℰ̂𝝂

01

𝑭 ′

0

1

0

1 𝒇

0

𝑭1


Since we have used LMN technique, we can

just say that is a good approximation of with

respect to uniform distribution. (though it is

also good with respect to , since we believe

in the conjecture that a Boolean formula like

outputs the same things most of the times

with respect to measure or )


However, this is enough for us, since we can

transform into by

And then using and playing with inequalities

-which is skipped for our present purpose-

will lead us to the proof.

FINALLY

By setting the parameter properly (i.e. ) one

can settle the conjecture:

Any independent distribution circuits of size

and depth :

REFERENCES [1] Linial, N., and Nisan, N. "Approximate inclusion exclusion." Combinatorica,

1990: 349- 365.

[2] Braverman, M. "Poly-logarithmic independence fools AC0 circuits." IEEE conference on Computational Complexity, 2009: 3-8.

[3] Bazzi, L. M. J. "Polylogarithmic independence can fool DNF formulas." Proceedings of the 48th annual IEEE symposium on Foundations of Computer Science, 2007: 63-73.

[4] Bazzi, L. M. J. "Polylogarithmic independence can fool DNF formulas." SIAM Journal on Computing (SICOMP), 2009.

[5] Razborov, A. A. "A simple proof of Bazzi's theorem." Electronic Colloquium on Computational Complexity, 2008: Report No. 81.

[6] Razborov, A. A. "Lower bounds for the size of circuits of bounded depth with basis {&,⊕}." Mathematicheskie Zametki, 1987: 598–607. English translation in Math. Notes. Acad.Sci. USSR, 1987: 333–338.

[7] Smolensky, R. "Algebraic methods in the theory of lower bounds for Boolean circuit complexity." Proceedings of the nineteenth annual ACM symposium on Theory of computing, 1987: 77-82.

[8] Linial, N., Mansour, Y. and Nisan, N. "Constant depth circuits, Fourier transform, and learnability." Journal of the ACM, 1993: 607-620.

[9] www.scottaaronson.com/blog/ retrieved on Nov 25th, 2010

http://www.scottaaronson.com/blog/

Poly-logarithmic independence fools circuits, a survey

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