1 2 Introduction We are going to use several consistency tests for Consistent Readers. We are going to use several consistency tests for Consistent Readers.Consistent

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IntroductionIntroduction

We are going to use several We are going to use several consistency tests for consistency tests for Consistent Consistent ReadersReaders..

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Plane Vs. Point Test - Plane Vs. Point Test - RepresentationRepresentation

RepresentationRepresentation:: One variable for each One variable for each planeplane pp of of

planes(planes()), supposedly assigned the , supposedly assigned the restriction of restriction of ƒƒ to to pp.. (Values of the (Values of the variables rang over all variables rang over all 22-dimensional, -dimensional, degree-degree-rr polynomials). polynomials).

One variable for eachOne variable for each pointpoint x x .. (Values of the variables rang(Values of the variables rang over the field over the field ).).

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Plane Vs. Point Test - TestPlane Vs. Point Test - Test

TestTest::

One local-test for every:One local-test for every:

planeplane pp and a and a pointpoint xx on on pp..

AcceptAccept if if AA’s value on ’s value on xx,, and and AA’s value on ’s value on p restricted to xp restricted to x are consistent. are consistent.

ReminderReminder::

AA: planes : planes dimension-2 degree-r polynomial dimension-2 degree-r polynomial

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Plane Vs. Point Test: Error Plane Vs. Point Test: Error ProbabilityProbability

ClaimClaim: :

The error probability of this test is The error probability of this test is very smallvery small, , i.e.i.e. < < c’/2 c’/2 , for some known , for some known 0<c’<10<c’<1..

The The error probabilityerror probability is the is the fractionfraction** of pairs of pairs <x, p><x, p> for for a a

point point xx and plane and plane pp whose: whose: AA’s value are ’s value are consistentconsistent, and yet , and yet Do Do notnot agree with any agree with any -permissible-permissible degree- degree-

rr polynomial (on the planes), polynomial (on the planes),

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

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

ProofProof: : By By reduction reduction to to Plane-Vs.-PlanePlane-Vs.-Plane testtest::

replace everyreplace every Local-test for Local-test for pp11 & & pp22 that that intersectintersect by a by a

line line ll, ,

by aby a Set of local-tests, one for each point Set of local-tests, one for each point xx on on

ll, that compares , that compares pp11’s & ’s & pp22’s values on ’s values on xx. .

Let’s denote this test by Let’s denote this test by PPx-TestPPx-TestWhat is its error-probability?What is its error-probability?

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

PropositionProposition:: The error-probability of The error-probability of PPx-TestPPx-Test is is ““almost the samealmost the same““ as as Plane-Vs.-PlanePlane-Vs.-Plane’s.’s.

ProofProof::The test errs in one of two cases:The test errs in one of two cases: First caseFirst case::

p1 p1 & & p2p2 agree on agree on ll, but, but Have Have impermissible impermissible values values (i.e. they do not (i.e. they do not

represent restrictions of 2 represent restrictions of 2 -permissible-permissible polynomials). polynomials).

Second caseSecond case:: p1 p1 & & p2p2 do not agreedo not agree on on ll, but, but Agree onAgree on the (randomly) chosen point the (randomly) chosen point xx on on

ll..

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In the first case In the first case Plane-Vs.-PlanePlane-Vs.-Plane also errs, so also errs, so according to according to [RaSa][RaSa], for some constant , for some constant 0<c<10<c<1 Pr(First-Case Error)Pr(First-Case Error) cc

For the second case, recall that:For the second case, recall that: rr == # #points, that two points, that two rr-degree, -degree, 11-dimensional -dimensional

polynomials can agree on.polynomials can agree on. |||| = = # #points on the line points on the line ll..

So So Pr(Second-Case Error) Pr(Second-Case Error) r/|r/|||

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

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For an appropriate For an appropriate (namely: (namely: ccO(r/|O(r/|

|)|)))::

c c + r/|+ r/|| = O(| = O(cc))

So, So, PPx-TestPPx-Test’s error-probability is ’s error-probability is

c’c’, for some , for some 0<c’<10<c’<1

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Back to Plane-Vs.-PointBack to Plane-Vs.-Point:: Let Let ppplanesplanes, , xx((pointspoints on on p p)), such that: , such that:

A(p)A(p) and and A(x)A(x) are are impermissibleimpermissible. . Let Let lllines lines such that such that xx l l Let Let p1p1,, p2 p2 be be planes planes throughthrough l l

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

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PrPrp, x p, x (( ((A(p)A(p)))(x) (x) = = A(x) A(x) ))

= Pr= Prllx, P1x, P1 ( ( ( (A(p1)A(p1)))(x) (x) = = A(x)A(x) ) )

==** E Ellxx ( Pr ( Prp1 p1 ( (( (A(p1)A(p1)))(x) (x) == A(x) A(x) || xxl l ) )) )

==**** E Ellxx ( ( (Pr(Prp1, p2 p1, p2 ( (( (A(p1)A(p1)))(x(x) = () = (A(p2)A(p2)))(x) (x) = = A(x) A(x) | | xxl l )) ) )1/21/2 ) )

(( E Ellxx (Pr(Prp1, p2 p1, p2 (( ((A(p1)A(p1)))(x) (x) = (= (A(p2)A(p2)))(x)(x) = = A(x) A(x) | | xxll ) ) ))1/21/2

** ( ( PrPrllx, p1, p2 x, p1, p2 (( ((A(p1)A(p1)))(x) (x) = (= (A(p2)A(p2)))(x) (x) = = A(x)A(x) ))1/21/2

****** ( (c’c’))1/21/2

** event A, and random variable Y, event A, and random variable Y, Pr(A) = EPr(A) = EYY( Pr(A|Y) )( Pr(A|Y) )** ** PrPrp1, p2 p1, p2 ( (A(p1))(x) = (A(p2))(x) = A(x) | x( (A(p1))(x) = (A(p2))(x) = A(x) | xL ) ) = L ) ) = ((p1,p2 p1,p2 are are independentindependent))

(Pr(Prp1 p1 ( (A(p1))(x) = A(x) | x( (A(p1))(x) = A(x) | xl ) )* (Prl ) )* (Prp1 p1 ( (A(p2))(x) = A(x) | x( (A(p2))(x) = A(x) | xl ) ) =l ) ) =

(Pr(Prp1 p1 ( (A(p1))(x) = A(x) | x( (A(p1))(x) = A(x) | xl ) )l ) )22

****** PPx-TestPPx-Test

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ConclusionConclusion::We’ve established that:We’ve established that:Plane-Vs.-PointPlane-Vs.-Point error probability, i.e., error probability, i.e.,The probability that The probability that pp (which is(which is randomrandom)) is is

Assigned an Assigned an impermissibleimpermissible value, and value, and This value This value agreesagrees with the value assigned with the value assigned

to to xx (which is also(which is also randomrandom),),

is is < < c’/2c’/2..

NoteNote: This proof is only valid as long as the point : This proof is only valid as long as the point xx whose value we would like to read is whose value we would like to read is randomrandom..

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Reading an Arbitrary Reading an Arbitrary PointPoint

Can we have similar Can we have similar procedureprocedure that that

would work for any would work for any arbitraryarbitrary point point xx??

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

returns an returns an impermissibleimpermissible value with only a value with only a smallsmall ((<<c’c’))

probabilityprobability..

Such procedure is called:Such procedure is called: consistent-readerconsistent-reader..

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Consistent Reader for Consistent Reader for Arbitrary PointArbitrary Point

RepresentationRepresentation: As in Plane-Vs-Point test.: As in Plane-Vs-Point test. local-readerslocal-readers: : InsteadInstead ofof local-testslocal-tests,, we we

have a set of (non Boolean) functions, have a set of (non Boolean) functions, [x] = [x] = {{11,...,,...,mm}}, referred to as: , referred to as: local-readerslocal-readers..

A local reader, can either A local reader, can either rejectreject or or return a return a valuevalue

from the field from the field ..

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

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33-Planes Consistent -Planes Consistent Reader for a Point xReader for a Point x

RepresentationRepresentation: One variable for each plane.: One variable for each plane.

Consistent-ReaderConsistent-Reader: :

For a point For a point xx, , [x][x] hashas one one local-readerlocal-reader [p[p22, p, p33]] for for

every pair of planes every pair of planes pp22 & p & p33 that intersects by a line that intersects by a line ll..

Let Let pp11 be the plane spanned by be the plane spanned by xx andand l l, , [p[p22, p, p33]]

rejectsrejects, unless , unless AA’s values on ’s values on pp11, , pp22 & & pp33 agree on agree on ll, ,

otherwiseotherwise: : returnsreturns AA’s value on ’s value on pp11 restricted to restricted to xx..

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Consistency ClaimConsistency ClaimClaimClaim:: WithWith highhigh probability probability ( ( 1- 1-c’c’))

RR [x][x] either rejects or either rejects or returnsreturns a permissible value a permissible value

for for xx..

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

RemarksRemarks::

The sign The sign RR is used for “is used for “randomly select from…randomly select from…”. ”.

Note that randomly selecting Note that randomly selecting XX and using it with and using it with ll to span to span PP11 is is

equal to randomly selecting equal to randomly selecting ll in in PP11 . .

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Consistency ProofConsistency Proof

ProofProof: : The value The value AA assigns assigns ll, according to , according to pp22 & &

pp33’s values, is ’s values, is permissiblepermissible w.h.p. w.h.p. (1-(1-c’c’))..

On the other hand, On the other hand, ll is a is a randomrandom line in line in pp11

and if and if pp11 is assigned an impermissible is assigned an impermissible

value value (by (by AA)), then that value restricted to , then that value restricted to

mostmost ll’s would be ’s would be impermissibleimpermissible..

with high probability

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Consistent-Reader for Consistent-Reader for Arbitrary Arbitrary kk points points

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

Let Let = {x = {x11, .., x, .., xkk}} be tuple of be tuple of kk point of the domain point of the domain ,, [ [ ] = { ] = { 11, .., , .., mm } } is now set of functions, which can is now set of functions, which can

either either rejectreject or or evaluateevaluate an an assignmentassignment to to xx11, .., x, .., xkk..

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

RepresentationRepresentation::

One One variablevariable for every cube (affine for every cube (affine

subspace) of dimension subspace) of dimension k+2k+2,, containingcontaining

..((Values of the variablesValues of the variables rang over all degree-rang over all degree-rr, ,

dimension dimension k+2k+2 polynomials ) polynomials )

one one variablevariable for every point for every point x x ..

(Values of the variables(Values of the variables rang over rang over ). ).

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

Show that the following distribution:Show that the following distribution: Choose a random cube Choose a random cube CC of dimension of dimension k+2k+2

containing containing Choose a random plane p in CChoose a random plane p in C Return pReturn p

Produces a distribution very close to Produces a distribution very close to uniform over planes puniform over planes p

Also, Also, pp w.h.p. does not contain a point of w.h.p. does not contain a point of ..

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Consistent Reader For Consistent Reader For kk ValuesValues - Cont. - Cont.

Consistent-ReaderConsistent-Reader:: OOnene local-reader local-reader for everyfor every cube cube C C containingcontaining

and a pointand a point y y C C, which, which

rejectsrejects ifif AA’s’s value for value for CC restricted to restricted to yy

disagreesdisagrees with with AA’s value on ’s value on y,y, otherwiseotherwise: : returnsreturns AA’s values on ’s values on C C restricted restricted

to to xx11, .., x, .., xk.k.

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Proof of ConsistencyProof of ConsistencyError Probability: Error Probability: c’/2c’/2

Suppose,Suppose, We have, in addition, a variable for each We have, in addition, a variable for each planeplane,, The test compares The test compares AA’s value on the ’s value on the cubecube CC

against against AA’s value on a ’s value on a planeplane pp, and then , and then against a against a pointpoint xx on that plane. on that plane.

The error probability doesn’t increaseThe error probability doesn’t increase..

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Proof of Consistency - Cont.Proof of Consistency - Cont.

PropositionProposition: This test induces a : This test induces a

distribution over the planes distribution over the planes pp which is which is

almost uniform.almost uniform.

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

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SummarySummary

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

Documents

1 2 Introduction We are going to use several consistency tests for Consistent Readers. We are going to use several consistency tests for Consistent Readers.Consistent