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Consistent Readers

Read Consistently a value for arbitrary points

Introduction

- We are going to use several consistency tests for Consistent Readers.

Plane Vs. Point Test - Representation

Representation:

- One variable for each planepof planes(), supposedly assigned the restriction of ƒ to p.(Values of the variables rang over all 2-dimensional, degree-r polynomials).
- One variable for eachpointx .(Values of the variables rangover the field ).

Plane Vs. Point Test - Test

Test:

- One local-test for every:

planepand a pointx on p.

- Accept if
- A’s value on x,and
- A’s value on p restricted to x are consistent.

Reminder:A: planes dimension-2 degree-r polynomial

Plane Vs. Point Test: Error Probability

Claim:

The error probability of this test is very small, i.e.< c’/2 , for some known 0

The error probability is the fraction*of pairs

point x and plane p whose:

- A’s value are consistent, and yet
- Do not agree with any -permissible degree-r polynomial (on the planes),

* fraction from the set of all combination of (point, plane)

Plane Vs. Point Test: Error Probability - Proof

Proof:

By reduction to Plane-Vs.-Planetest:

replace every

- Local-test for p1 & p2 that intersect by a line l,

by a

- Set of local-tests, one for each point x on l, that compares p1’s & p2’s values on x.

Let’s denote this test by PPx-Test

What is its error-probability?

Plane Vs. Point Test: Error Probability - Proof Cont.

Proposition: The error-probability of PPx-Test is “almost the same“as Plane-Vs.-Plane’s.

Proof:

The test errs in one of two cases:

- First case:
- p1& p2agree on l, but
- Have impermissible values (i.e. they do not represent restrictions of 2 -permissiblepolynomials).
- Second case:
- p1& p2do not agreeon l, but
- Agree on the (randomly) chosen point x on l.

Plane Vs. Point Test: Error Probability - Proof Cont.

- In the first case Plane-Vs.-Plane also errs, so according to [RaSa], for some constant 0
- For the second case, recall that:
- r= #points, that two r-degree, 1-dimensional polynomials can agree on.
- || = #points on the line l.

So Pr(Second-Case Error) £ r/||

PPx-Test’s error-probability £ c + r/||

Plane Vs. Point Test: Error Probability - Proof Cont.

For an appropriate (namely: cO(r/||)):

c + r/|| = O(c)

So, PPx-Test’s error-probability is

£ c’, for some 0

Plane Vs. Point Test: Error Probability - Proof Cont.

Back to Plane-Vs.-Point:

- Let pplanes, x(points on p), such that:
- A(p) and A(x) are impermissible.
- Let llines such that x l
- Let p1, p2 be planes through l

Plane-Vs.-Point’s error probability is:

Pr p, x ((A(p))(x) = A(x) ) =

= Pr lx, p1 ( (A(p1))(x) = A(x) )

Plane Vs. Point Test: Error Probability - Proof Cont.

Prp, x ((A(p))(x) = A(x) )

= Prlx, P1 ( (A(p1))(x) = A(x) )

=* Elx ( Prp1 ( (A(p1))(x) = A(x)|xl ) )

=**Elx( (Prp1, p2 ( (A(p1))(x) = (A(p2))(x) = A(x) | xl ) )1/2 )

( Elx(Prp1, p2 ((A(p1))(x) = (A(p2))(x) = A(x) | xl ) )1/2

*( Prlx, p1, p2 ((A(p1))(x) = (A(p2))(x) = A(x))1/2

*** (c’)1/2

*event A, and random variable Y, Pr(A) = EY( Pr(A|Y) )

** Prp1, p2 ( (A(p1))(x) = (A(p2))(x) = A(x) | xL ) ) = (p1,p2 are independent)

(Prp1 ( (A(p1))(x) = A(x) | xl ) )* (Prp1 ( (A(p2))(x) = A(x) | xl ) ) =

(Prp1 ( (A(p1))(x) = A(x) | xl ) )2

***PPx-Test

Plane Vs. Point Test: Error Probability - Proof Cont.

Conclusion:

We’ve established that:

Plane-Vs.-Point error probability, i.e.,

The probability that p(which israndom)is

- Assigned an impermissible value, and
- This value agrees with the value assigned to x(which is alsorandom),

is < c’/2.

Note: This proof is only valid as long as the point x whose value we would like to read is random.

Reading an Arbitrary Point

Can we have similar procedure that

would work for any arbitrary point x?

i.e., a set of evaluating functions, where the function

returns an impermissible value with only a small(<c’)

probability.

Such procedure is called:consistent-reader.

Consistent Reader for Arbitrary Point

- Representation: As in Plane-Vs-Point test.
- local-readers: Insteadoflocal-tests,we have a set of (non Boolean) functions, [x] = {1,...,m}, referred to as: local-readers.

A local reader, can either reject or return a value

from the field .

[supposedly the value is ƒ(x), with ƒ a degree-r polynomial].

3-Planes Consistent Reader for a Point x

Representation: One variable for each plane.

Consistent-Reader:

- For a point x, [x]hasone local-reader[p2, p3] for every pair of planes p2 & p3 that intersect by a line l.
- Let p1 be the plane spanned byxand l, [p2, p3]
- rejects, unless A’s values on p1, p2 & p3agree on l,
- otherwise: returnsA’s value on p1 restricted to x.

Consistency Claim

Claim:Withhighprobability ( 1-c’) R [x]either rejects or returns a permissible value for x.

[i.e., consistent with one of the permissible polynomials].

Remarks:

- The sign Ris used for “randomly select from…”.
- Note that randomly selecting X and using it with l to spanp1is equal to randomly selecting l in p1.

with high probabilityConsistency Proof

Proof:

- The value A assigns l, according to p2 & p3’s values, is permissible w.h.p. (1-c’).
- On the other hand, lis a random line in p1 and if p1 is assigned an impermissible value (by A), then that value restricted to mostl’s would be impermissible.

Consistent-Reader for Arbitrary k points

How can we read consistently more than one value ?

Note: Using the point-consistent-reader, we need to invoke the reader several times, and the received values may correspond to differentpermissible polynomials.

- Let = {x1, .., xk} be tuple of k point of the domain ,
- [ ] = { 1, .., m } is now set of functions, which can either reject or evaluate an assignment tox1, .., xk.

Hyper-Cube-Vs.-Point Consistent-Reader For k Points

Representation:

- One variable for every cube (affine subspace) of dimension k+2,containing.(Values of the variablesrang over all degree-r, dimension k+2 polynomials )
- one variable for every point x .(Values of the variablesrang over ).

Hyper-Cube-Vs.-Point Consistent-Reader For k Points

- Show that the following distribution:
- Choose a random cube C of dimension k+2containing
- Choose a random plane p in C
- Return p

Produces a distribution very close to uniform over planes pAlso, p w.h.p. does not contain a point of .

Consistent Reader For k Values - Cont.

Consistent-Reader:

- One local-reader for every cube C containingand a pointy C, which
- rejectsifA’svalue for C restricted to ydisagrees with A’s value on y,
- otherwise: returnsA’s values on C restricted to x1, .., xk.

Proof of Consistency

Error Probability: c’/2

Suppose,

- We have, in addition, a variable for each plane,
- The test compares A’s value on the cubeC
- against A’s value on a planep, and then
- against a pointx on that plane.

The error probability doesn’t increase.

Proof of Consistency - Cont.

Proposition: This test induces a distribution over the planes p which is almost uniform.

Lemma: Plane-Vs.-Point test works the same if instead of assigning a single value, one assigns each plane with a distribution over values.

Summary

- We saw some consistent readersand how “accurate” they are. They will be a useful tool in this proof.

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