11

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IntroductionIntroduction In this chapter we examine In this chapter we examine consistency testsconsistency tests, and , and

trying to improve their parameters: trying to improve their parameters: reducing the number of variables accessed by reducing the number of variables accessed by

the test.the test. reducing the variables’ range.reducing the variables’ range. reducing error probabilityreducing error probability..

We present the tests: We present the tests: Points-on-LinePoints-on-Line Line-vs.-PointLine-vs.-Point Plane-vs.-PlanePlane-vs.-Plane

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Basic TermsBasic Terms

The Basic Terms:The Basic Terms:

RepresentationRepresentation [.][.] [.][.] is a set of variables, for which is a set of variables, for which

a value is assigned,a value is assigned, The values are in the range 2The values are in the range 2vv,, The values correspond to a single, The values correspond to a single,

polynomial polynomial ƒ: ƒ: of global of global degree degree rr

V from PCP[D, V, )

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Basic TermsBasic Terms

TestTest A set of Boolean functions (local A set of Boolean functions (local

tests) tests) Each depends on at most Each depends on at most DD

representation’s variables. representation’s variables.

D from PCP[D, V, )

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Basic TermsBasic Terms

ConsistencyConsistency: : Measures an amount of Measures an amount of

conformation between the conformation between the different values assigned to the different values assigned to the representation variables.representation variables.

We say that the values are We say that the values are consistent if they satisfy at least consistent if they satisfy at least an an -fraction of the local tests.-fraction of the local tests.

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Affine subspacesAffine subspaces

Let us define some specific affine Let us define some specific affine subspaces ofsubspaces of::

lines(lines()) is the set of all lines (affine is the set of all lines (affine subspaces of dimension subspaces of dimension 11) of) of

planes(planes()) is the set of all planes is the set of all planes (affine subspaces of dimension (affine subspaces of dimension 22) of) of

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Overview of the TestsOverview of the Tests In each tests the variables in In each tests the variables in [.][.]

represent some aspect of the given represent some aspect of the given polynomial polynomial ff, such as, such as

ff’s values on points of ’s values on points of ff’s restriction to a line in ’s restriction to a line in ff’s restriction to a plane in ’s restriction to a plane in

The local-tests check compatibility The local-tests check compatibility between the values of different between the values of different variables in variables in [.][.]..

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Simple Test: Points-on-Simple Test: Points-on-LineLine

RepresentationRepresentation:: [.][.] has one variable has one variable [p][p] for each for each pointpoint

pp. . The variables are supposedly assigned The variables are supposedly assigned

the valuethe value ƒ(p)ƒ(p) Hence the range of the variables is: Hence the range of the variables is:

v = log ||

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Points-on-Line: TestPoints-on-Line: Test

TestTest:: There’s one local-test for each There’s one local-test for each lineline lllines(lines()).. Each test depends on all points ofEach test depends on all points of l l (altogether (altogether

2r2r points). points).

A testA test acceptsaccepts if and only if the values are if and only if the values are

consistent with a single degree-consistent with a single degree-rr univariate univariate

polynomialpolynomial

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Points-on-Line: Points-on-Line: ConsistencyConsistency

DefDef: An assignment to : An assignment to is said to be is said to be globally globally

consistentconsistent if values on if values on mostmost points agree points agree

with awith a single single, global degree-, global degree-rr polynomial. polynomial.

ThmThm[RuSu]:[RuSu]: If a large (constant) fraction of If a large (constant) fraction of

the local-tests accept, then there is a the local-tests accept, then there is a

polynomial polynomial ƒ ƒ ((of of degree-degree-rr) which agrees ) which agrees

with the assigned values on most points. with the assigned values on most points.

Alas, Alas, each local-test each local-test depends on adepends on a

non constantnon constant number of variablesnumber of variables

(2r)(2r)

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Next Test: Line-vs.-PointNext Test: Line-vs.-Point

RepresentationRepresentation:: [.][.] has one variable has one variable [p][p] for each for each

pointpoint pp, supposedly assigned, supposedly assigned ƒ(p)ƒ(p),,

Plus, one variable Plus, one variable [l][l] for each for each lineline lllines(lines()),, supposedly assignedsupposedly assigned ƒƒ ’s’s restriction torestriction to ll..

Hence the range of [l] is all degree-r univariate poly’s

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Line-vs.-Point: TestLine-vs.-Point: Test

TestTest:: There’sThere’s one local-test for each pair of:one local-test for each pair of:

a line a line l l lines( lines()), and , and a point a point p p l l ..

A testA test accepts accepts if the value assigned to if the value assigned to [p][p] equals the value of the equals the value of the polynomial assigned to polynomial assigned to [l][l] on the on the point point pp..

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Global Consistency: Global Consistency: Constant ErrorConstant Error

ThmThm [AS,ALMSS]: [AS,ALMSS]: Probability of finding Probability of finding inconsistency, between value for inconsistency, between value for [p][p] and and value for line value for line [l][l] on on pp, is high (constant) ,, is high (constant) ,

unlessunless

mostmost lines and most points agree with a lines and most points agree with a single, global degree-single, global degree-rr polynomial.polynomial.

HereHere D = O(1) V = (r+1) log|D = O(1) V = (r+1) log||| & & constant constant..

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Can the Test Be Improved?Can the Test Be Improved?

Can error-probability be made smaller than Can error-probability be made smaller than constant (such as constant (such as 1/log(n)1/log(n) ), while keeping ), while keeping each local-test depending on constant each local-test depending on constant number of representation variables?number of representation variables?

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What’s the problem?What’s the problem?

Adversary: randomly partition variables into Adversary: randomly partition variables into kk sets, each consistent with a distinct sets, each consistent with a distinct degree-degree-rr polynomial polynomialThis would cause the local-test’s success This would cause the local-test’s success probability to be at least probability to be at least kk-(D-1)-(D-1)..

(if all variables fall within the same set in the (if all variables fall within the same set in the partition)partition)

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ConsequentlyConsequently

One therefore must further One therefore must further weakenweaken the the notion of notion of global consistencyglobal consistency sought sought afterafter

[ [ still, making sure it can be applied in still, making sure it can be applied in order to deduceorder to deduce PCPPCP characterization ofcharacterization of NPNP ].].

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Limited PluralismLimited Pluralism

DefDef: Given an assignment to : Given an assignment to ’s variables,’s variables,a degree-a degree-rr polynomial polynomial ƒƒ is said to be is said to be--permissiblepermissible if it is consistent with at least if it is consistent with at least a a fraction of the values assigned. fraction of the values assigned.

Global ConsistencyGlobal Consistency: assignment’s values : assignment’s values consistent with any consistent with any -permissible ƒ-permissible ƒ are are acceptable. acceptable.

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Limited Pluralism - Cont.Limited Pluralism - Cont.

Formally:Formally:

DefDef: A local test is said to : A local test is said to errerr (with respect (with respect to to ) if it ) if it acceptsaccepts values that are values that are NOT NOT consistentconsistent with with anyany -permissible-permissible degree- degree-rr ƒƒ ’s.’s.

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Limited Pluralism - Cont.Limited Pluralism - Cont.

Note that the adversary’s randomly Note that the adversary’s randomly partition does not trick the test this time:partition does not trick the test this time:

If the test accepts when all the variables If the test accepts when all the variables are from a set consistent with an r-degree are from a set consistent with an r-degree polynomial, then the polynomial is really polynomial, then the polynomial is really --permissible.permissible.

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Plane-vs.-Plane: Plane-vs.-Plane: RepresentationRepresentation

RepresentationRepresentation:: [.][.] has one variable has one variable [p][p] for each for each

planeplane ppplanes(planes()), , supposedlysupposedly assignedassigned the the

restriction of restriction of ff to to pp..

Hence the range of [p] is all degree-r two-variables poly’s

2121

Plane-vs.-Plane: Plane-vs.-Plane: RepresentationRepresentation

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Plane-vs.-Plane: TestPlane-vs.-Plane: Test

TestTest:: There’s one local-test for each line There’s one local-test for each line

lllines(lines()) and a pair of planes and a pair of planes pp11,p,p22planes(planes()) such that such that llpp11 and and llpp22

A testA test accepts accepts if and only if the value ofif and only if the value of [p[p11]] restricted torestricted to l l equals the value ofequals the value of

[p[p22]] restricted torestricted to l l..

HereHere D=O(1), v=2(r+1)D=O(1), v=2(r+1)22log|log||.|.

That is, a pair

of plains

intersecting

by a line

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Plane-vs.-Plane: Plane-vs.-Plane: Consistency Consistency

ThmThm[RaSa]:[RaSa]:

As long as As long as ||||-c-c for some constant for some constant

1 > c > 01 > c > 0, a local test err (w.r.t. , a local test err (w.r.t. ) )

with a very small probability, namely with a very small probability, namely

c’c’ for some constant for some constant 1 > c’ > 01 > c’ > 0..

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Plane-vs.-Plane: Plane-vs.-Plane: Consistency - Cont.Consistency - Cont.

The theorem states that, the plane-vs.-plane The theorem states that, the plane-vs.-plane

test, with very high probability test, with very high probability

(( 1 - 1 - c’c’), either ), either rejectsrejects, or , or acceptsaccepts values of values of

a a -permissible-permissible polynomial polynomial ..

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SummarySummary

We examined consistency tests, We examined consistency tests, Points-on-Line,Line-vs.-Point and Points-on-Line,Line-vs.-Point and Plane-vs.-Plane.Plane-vs.-Plane.

By weakening toBy weakening to -permissible-permissible

definition, we achieve an error definition, we achieve an error probability which is below constantprobability which is below constant..