Download - Consistent Readers
![Page 1: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/1.jpg)
1
![Page 2: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/2.jpg)
2
IntroductionIntroduction
We are going to use several consistency tests for Consistent Readers.
![Page 3: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/3.jpg)
3
Plane Vs. Point Test - Plane Vs. Point Test - RepresentationRepresentation
RepresentationRepresentation:One variable for each planeplane pp of
planes(), supposedly assigned the restriction of ƒƒ to p. (Values of the variables rang over all 2-dimensional, degree-r polynomials).
One variable for each pointpoint xx . (Values of the variables rang over the field ).
![Page 4: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/4.jpg)
4
Plane Vs. Point Test - TestPlane Vs. Point Test - Test
TestTest:
One local-test for every:
planeplane pp and a pointpoint xx on p.
AcceptAccept if – A’s value on x, and
– A’s value on p restricted to x are consistent.
Reminder:
AA: planes dimension-2 degree-r polynomial
![Page 5: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/5.jpg)
5
Plane Vs. Point Test: Error Plane Vs. Point Test: Error ProbabilityProbability
ClaimClaim: The error probability of this test is very small,
i.e. < c’/2 , for some known 0<c’<1.
The error probability is the fraction* of pairs <x, p> for a
point x and plane p whose: – A’s value are consistent, and yet – Do not agree with any -permissible-permissible degree-r
polynomial (on the planes),
* fraction from the set of all combination of (point, plane)
![Page 6: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/6.jpg)
6
Plane Vs. Point Test: Error Plane Vs. Point Test: Error Probability - ProofProbability - Proof
ProofProof: By reduction to Plane-Vs.-Plane test: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-TestPPx-TestWhat is its error-probability?
![Page 7: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/7.jpg)
7
Plane Vs. Point Test: Error Plane Vs. Point Test: Error Probability - Proof Cont.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 & p2 agree on l, but– Have impermissible values (i.e. they do not
represent restrictions of 2 -permissible polynomials).
Second case:– p1 & p2 do not agree on l, but – Agree on the (randomly) chosen point x on l.
![Page 8: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/8.jpg)
8
Plane Vs. Point Test: Error Plane Vs. Point Test: Error Probability - Proof Cont.Probability - Proof Cont.
In the first case Plane-Vs.-Plane also errs, so according to [RaSa], for some constant 0<c<1 Pr(First-Case Error)Pr(First-Case Error) cc
For the second case, recall that:– rr = #points, that two r-degree, 1-dimensional
polynomials can agree on.
– |||| = #points on the line l.
So Pr(Second-Case Error) Pr(Second-Case Error) r/|r/|||
PPx-Test’s error-probability c c + r/|+ r/|||
![Page 9: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/9.jpg)
9
Plane Vs. Point Test: Error Plane Vs. Point Test: Error Probability - Proof Cont.Probability - Proof Cont.
For an appropriate (namely: (namely: ccO(r/|O(r/|
|)|)))::
c c + r/|+ r/|| = O(| = O(cc))
So, PPx-Test’s error-probability is
c’c’, for some 0<c’<1
![Page 10: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/10.jpg)
10
Plane Vs. Point Test: Error Plane Vs. Point Test: Error Probability - Proof Cont.Probability - Proof Cont.Back to Plane-Vs.-PointBack to Plane-Vs.-Point:: Let ppplanesplanes, xx((pointspoints on on p p)), such that:
– A(p)A(p) and A(x)A(x) are impermissible. Let lllines lines such that x l Let p1, p2 be planes through l
Plane-Vs.-Point’s error probability is:
Pr Pr p, x p, x (( ((A(p)A(p)))(x) (x) = = A(x) A(x) ) =) =
= Pr Pr llx, p1 x, p1 ( (( (A(p1)A(p1)))(x) (x) = = A(x)A(x) ) )
![Page 11: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/11.jpg)
11
Plane Vs. Point Test: Error Plane Vs. Point Test: Error Probability - Proof Cont.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’c’)1/21/2
** 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 ) )22
****** PPx-Test
![Page 12: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/12.jpg)
12
Plane Vs. Point Test: Error Plane Vs. Point Test: Error Probability - Proof Cont.Probability - Proof Cont.ConclusionConclusion::We’ve established that:Plane-Vs.-Point error probability, i.e.,The probability that p (which is random) is
– Assigned an impermissible value, and– This value agrees with the value assigned to x
(which is also random),
is < < c’/2c’/2.
Note: This proof is only valid as long as the point x whose value we would like to read is randomrandom.
![Page 13: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/13.jpg)
13
Reading an Arbitrary PointReading an Arbitrary Point
Can we have similar procedure that
would work for any arbitraryarbitrary 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-readerconsistent-reader..
![Page 14: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/14.jpg)
14
Consistent Reader for Consistent Reader for Arbitrary Arbitrary PointPoint
Representation: As in Plane-Vs-Point test.local-readerslocal-readers: Instead of local-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].
![Page 15: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/15.jpg)
15
33-Planes Consistent Reader -Planes Consistent Reader for a Point for a Point xx
Representation: One variable for each plane.
Consistent-Reader:
For a point x, [x] has one local-reader [p2, p3]
for every pair of planes p2 & p3 that intersect by a
line l.
Let p1 be the plane spanned by x and l, [p2, p3]
– rejects, unless A’s values on p1, p2 & p3 agree on l,
– otherwise: returns A’s value on p1 restricted to x.
![Page 16: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/16.jpg)
16
Consistency ClaimConsistency Claim
Claim: With high probability ( 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 R is used for “randomly select from…”.
Note that randomly selecting X and using it with l to span p1 is
equal to randomly selecting l in p1.
![Page 17: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/17.jpg)
17
Consistency ProofConsistency Proof
Proof: The value A assigns l, according to p2 &
p3’s values, is permissible w.h.p. (1-c’).
On the other hand, l is a random line in
p1 and if p1 is assigned an impermissible
value (by A), then that value restricted to most l’s would be impermissible.
with high probability
![Page 18: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/18.jpg)
18
Consistent-Reader for Arbitrary Consistent-Reader for Arbitrary kk pointspoints
How can we read consistently How can we read consistently more more than one value than one value ??
Note: Using the point-consistent-reader, we need to invoke the reader several times, and the received values may correspond to different permissible 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 to x1, .., xk.
![Page 19: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/19.jpg)
19
Hyper-Cube-Vs.-Point Hyper-Cube-Vs.-Point Consistent-Reader For Consistent-Reader For kk Points Points
Representation:
One variable for every cube (affine subspace) of dimension k+2, containing .(Values of the variables rang over all degree-r, dimension k+2 polynomials )
one variable for every point x .
(Values of the variables rang over ).
![Page 20: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/20.jpg)
20
Hyper-Cube-Vs.-Point Hyper-Cube-Vs.-Point Consistent-Reader For Consistent-Reader For kk Points Points
Show that the following distribution:– Choose a random cube C of dimension
k+2 containing – Choose a random plane p in C– Return p
Produces a distribution very close to uniform over planes p
Also, p w.h.p. does not contain a point of .
![Page 21: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/21.jpg)
21
Consistent Reader For Consistent Reader For kk Values Values - - Cont.Cont.
Consistent-Reader:
One local-reader for every cube C containing
and a point y C, which
– rejects if A’s value for C restricted to y disagrees with A’s value on y,
– otherwise: returns A’s values on C
restricted to x1, .., xk.
![Page 22: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/22.jpg)
22
Proof of ConsistencyProof of Consistency
Error Probability: c’/2
Suppose, We have, in addition, a variable for each
plane, The test compares A’s value on the cube C
– against A’s value on a plane p, and then
– against a point x on that plane.
The error probability doesn’t increase.
![Page 23: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/23.jpg)
23
Proof of Consistency - Cont.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.
![Page 24: Consistent Readers](https://reader035.vdocuments.us/reader035/viewer/2022062520/56815949550346895dc683c7/html5/thumbnails/24.jpg)
24
SummarySummary
We saw some consistent readers and how “accurate” they are. They will be a useful tool in this proof.