1 Introduction l In this chapter we examine

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Introduction l In this chapter we examine consistency tests, and trying to improve their parameters: – reducing the number of variables accessed by the test. – reducing the variables’ range. – reducing error probability. 2

Introduction We present the tests: l Points-on-Line l Line-vs. -Point l Plane-vs. -Plane 3

Representation, Test, Consistency The Basic Terms: l V from PCP[D, V, ) Representation [. ] – [. ] is a set of variables, – To each variable a value is assigned, – The values are in the range 2 v, – The values correspond to a single, polynomial ƒ: a f is of global degree r 4

Representation, Test, Consistency D from PCP[D, V, ) l Test – A set of Boolean functions, – Each depends on at most D representation’s variables. local tests 5

Representation, Test, Consistency l Consistency: Consistency – Measures an amount of conformation between the different values assigned to the representation variables. – We say that the values are consistent if they satisfy at least an -fraction of the local tests. 6

Geometry l Let us define some specific affine subspaces of : – lines( ) is the set of all lines (affine subspaces of dimension 1) of – planes( ) is the set of all planes (affine subspaces of dimension 2) of 7

Overview of the Tests l In each tests the variables in [. ] represent some aspect of the given polynomial f, such as – f’s values on points of – f’s restriction to a line in – f’s restriction to a plane in l The local-tests check compatibility between the values of different variables in [. ]. 8

Simple Test: Points-on-Line Representation: Representation has one variable [p] for each point p . l [. ] l The variables are supposedly assigned the value ƒ(p). hence v = log | | 9

Points-on-Line: Test: Test l There’s one local-test for each line l lines( ). l Each 2 r test depends on all points of l. A test accepts if and only if the values are consistent with a single degree-r univariate polynomial 10

Points-on-Line: Consistency Alas, Def: An assignment to iseach saidlocal-test to be depends on a globally consistentnon if values on most constant points agree with a single, global degree-r number of variables (2 r) polynomial. Thm[Ru. Su]: If a large (constant) fraction of the local-tests accept, then there is a polynomial ƒ (of degree-r) which agrees with the assigned values on most points. 11

Next Test: Line-vs. -Point Representation: Representation has one variable [p] for each point p , supposedly assigned ƒ(p), l [. ] l Plus, one variable [l] for each line l lines( ), supposedly assigned ƒ ’s restriction to l. Hence the range of [l] is all degree-r univariate poly’s 12

Line-vs. -Point: Test: Test l There’s of: one local-test for each pair – a line l lines( ), and – a point p l. l. A test accepts if the value assigned to [p] equals the value of the polynomial assigned to [l] on the point p. 13

Global Consistency: Constant Error Thm [AS, ALMSS]: Probability of finding inconsistency, between value for [p] and value for line [l] on p, is high (constant) , unless most lines and most points agree with a single, global degree-r polynomial. Here D = O(1) V = (r+1) log| | & constant. 14

Can the Test Be Improved? Can error-probability be made smaller than constant (such as 1/log(n) ), while keeping each local-test depending on constant number of representation variables? 15

What’s the problem? Adversary: randomly partition variables into k sets, each consistent with a distinct degree-r polynomial This would cause the local-test’s success probability to be at least k-(D-1). (if all variables fall within the same set in the partition) 16

Consequently One therefore must further weaken the notion of global consistency sought after [ still, making sure it can be applied in order to deduce PCP characterization of NP ]. 17

Limited Pluralism Def: Given an assignment to ’s variables, a degree-r polynomial ƒ is said to be -permissible if it is consistent with at least a fraction of the values assigned. Global Consistency: assignment’s values consistent with any -permissible ƒ are acceptable. 18

Limited Pluralism - Cont. Formally: Def: A local test is said to err (with respect to ) if it accepts values that are NOT consistent with any -permissible degree-r ƒ ’s. 19

Limited Pluralism - Cont. l Note that the adversary’s randomly partition does not trick the test this time: l If the test accepts when all the variables are from a set consistent with an r-degree polynomial, then the polynomial is really permissible. 20

Plane-vs. -Plane: Representation: l [. ] has one variable [p] for each plane p planes( ), l supposedly assigned the restriction of f to p. Hence the range of [p] is all degree-r two-variables poly’s 21

Plane-vs. -Plane: Representation 22

Plane-vs. -Plane: Test: That is, a pair of plains intersecting by a line l There’s one local-test for each line l lines( ) and a pair of planes p 1, p 2 planes( ) such that l p 1 and l p 2 l. A test accepts if and only if the value of [p 1] restricted to l equals the value of [p 2] restricted to l. Here D=O(1), v=2(r+1)2 log| |. 23

Plane-vs. -Plane: Consistency Thm[Ra. Sa]: As long as ³| |-c for some constant 1 > c > 0, a local test err (w. r. t. ) with a very small probability, namely £ c’ for some constant 1 > c’ > 0. 24

Plane-vs. -Plane: Consistency - Cont. The theorem states that, the plane-vs. plane test, with very high probability (³ 1 - c’), either rejects, or accepts values of a -permissible polynomial . 25

Summary l We examined consistency tests, Points-on-Line, Line-vs. -Point and Plane -vs. -Plane. l By weakening to -permissible definition, we achieve an error probability which is below constant. 26