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Building The Ultimate Consistent Reader

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IntroductionIntroduction

We’ve already built a consistent reader (cube-Vs.-point)...

Except it had variables ranging over a set of polynomials instead of over the field.

In this lecture we’ll use that construction to build a perfected consistent reader.

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Starting Point: Cube-Vs.-PointStarting Point: Cube-Vs.-Point

cube var.

point var. ?

supposedly assigned the restriction of the polynomial to that

cube

supposedly assigned the value of the polynomial in that point

Each cube is actually a

new domain!

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General Framework: SketchGeneral Framework: Sketch

consistent reader

“better” consistent

reader

extension

composition

repeat this process

replacing the cube variables by variables with lower

degree

adding new readers

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ExtensionsExtensions

We’ll introduce two extension procedures:– Power Substitution– Linearization Extension

Both embed the domains in new domains with higher dimension, but much lower degree.

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ExtensionsExtensions

All our transformations, but the last one, will use the power substitution extension procedure.

The final reader will be created using the linearization extension.

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Power Substitution - Example Power Substitution - Example (b=3)(b=3)

x31 + x15 = 0

(31)3=1011

(15)3=120

i=0,1,... xi:=x3i

x0x1x3 + x12x2 =

0

How many new

variables do we need at

most? t:=log3(s+1)

Bound the degree: 2t

Extend this idea to general b (base) and d (dimension)

Now, say the total degree of the polynomial deg satisfies

rdegs.

deg

r/3t

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Embedding ExtensionEmbedding Extension

d

dt

(x1,...,xd)(x1

b0,..,x1

bt,....,xd

bt)

Apply this transformation to every cube, where

(the parameter s associated with the domain)

}2,)1max{(1log

1

nsb

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Composition: Composition: How To Build A How To Build A Consistent Reader For The New Domains?Consistent Reader For The New Domains?

For each local reader (having variables from exactly one domain) generate a Cube-Vs.-Point reader.

Replace each occurrence of these variables in the original reader with the proper evaluation.

Put the new local test in conjunction with the existing local test.

verify this procedure takes polynomial time

Make sure you can prove correctness.

We only need to read one point! Hence the dimension of the new domains is constant

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What Next?What Next?

Repeat this process until

nd

dsnew

newnew

1log

When will this happen? Recall that b=(s+1)1/log1-n (as long as this

is not less than 2) t=log1-n. snew=dt(b-1)=polylog(n)s1/log1-n Thus in the i’th iteration, s=2O(log-i(1-)n)

(when -i(1-)>0).

(*)

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Degree Decreases RapidlyDegree Decreases Rapidly

Let i0:=/(1-).

1--(i0-1)(1-)>0

Hence in the (i0-1)’th iteration,

s=2O(log-(i0-1)(1-)n).

Then b=O(1).

Consequently, t=O(log1-n).

In the (i0+1)’th iteration (*) should hold,

since the dimension is constant.

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Obtaining Linear PolynomialsObtaining Linear Polynomials

When (s+d

d) is small enough, we can apply another technique, called linearization extension, to obtain linear polynomials.

Which means our consistent reader relies on constant number of representation variables.

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Linerization - ExampleLinerization - Example

x2yz + xy2 + z = 0

ux2yz + uxy2 + uz = 0

How many new

variables do we need at

most? ux2yz := x2yz uxy2 := xy2

uz := z

Now, say the total degree of the polynomial deg satisfies rdegs. The

dimension of the polynomial is d.

d

dsM :

Linear polynomi

al!

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Linearization EmbeddingLinearization Embedding

d

M

(x1,...,xd) (m1(x1,..,xd),..,mM(x1,..))

Apply this transformation to every cube (where m1,...,mM are all the degree-s

dimension-d monomials):

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SummarySummary

Using the Cube-Vs.-Point consistent reader as a black box, we’ve managed to build an adequate consistent reader:– Each local reader depends on

constant number of variables.– All variables range over the field.– The error probability is small.

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AppendixAppendix

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Linearization: How Many Linearization: How Many Degree-s, Dimension-d Degree-s, Dimension-d Monomials?Monomials?Or equivalently - how many partitions of at most s balls into d

boxes?

+1

. .

.s balls

d mobile partitions

d

ds

ds

ds

!!

)!(d identical

partitions, s identical balls

s+d objects arranged in a row