1 Introduction n In this chapter we examine

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Introduction n In this chapter we examine consistency tests, and trying to improve their parameters: n reducing the number of variables accessed by the test. n reducing the variables’ range. n reducing error probability. We present the tests: n Points-on-Line n Line-vs. -Point n Plane-vs. -Plane 2

Basic Terms The Basic Terms: n V from PCP[D, V, ) Representation [. ] is a set of variables, for which a value is assigned, n The values are in the range 2 v, n The values correspond to a single, polynomial ƒ: a of global degree r 3

Basic Terms n Test n A set of Boolean functions (local tests) n Each depends on at most D representation’s variables. D from PCP[D, V, ) 4

Basic Terms n Consistency: n Measures an amount of conformation between the different values assigned to the representation variables. n We say that the values are consistent if they satisfy at least an -fraction of the local tests. 5

Affine subspaces n Let us define some specific affine subspaces of : n n 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 6

Overview of the Tests n n In each tests the variables in [. ] represent some aspect of the given polynomial f, such as n f’s values on points of n f’s restriction to a line in n f’s restriction to a plane in The local-tests check compatibility between the values of different variables in [. ]. 7

Simple Test: Points-on-Line Representation: n n n [. ] has one variable [p] for each point p . The variables are supposedly assigned the value ƒ(p) Hence the range of the variables is: v = log | | 8

Points-on-Line: Test: n n n There’s one local-test for each line l lines( ). Each test depends on all points of l (altogether 2 r points). A test accepts if and only if the values are consistent with a single degree-r univariate polynomial 9

Points-on-Line: Consistency Def: An assignment to Alas, is said to be globally each consistent if values on local-test most points agree with a single, globaldepends degree-onr apolynomial. non constant number of variables (2 r) 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. 10

Next Test: Line-vs. -Point Representation: n n [. ] has one variable [p] for each point p , supposedly assigned ƒ(p), 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 11

Line-vs. -Point: Test: n n There’s one local-test for each pair of: n a line l lines( ), and n a point p l. A test accepts if the value assigned to [p] equals the value of the polynomial assigned to [l] on the point p. 12

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. 13

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? 14

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) 15

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 ]. 16

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. 17

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. 18

Limited Pluralism - Cont. n n Note that the adversary’s randomly partition does not trick the test this time: If the test accepts when all the variables are from a set consistent with an r-degree polynomial, then the polynomial is really permissible. 19

Plane-vs. -Plane: Representation: n [. ] has one variable [p] for each plane p planes( ), n supposedly assigned the restriction of f to p. Hence the range of [p] is all degree-r two-variables poly’s 20

Plane-vs. -Plane: Representation 21

That is, a pair of plains intersecting by a line Plane-vs. -Plane: Test: n n 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 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| |. 22

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. 23

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. 24

Summary n n We examined consistency tests, Pointson-Line, Line-vs. -Point and Plane-vs. -Plane. By weakening to -permissible definition, we achieve an error probability which is below constant. 25