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3. 2 Introduction 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.

4. 3 Introduction We present the tests: Points-on-Line Line-vs.-Point Plane-vs.-Plane

5. 4 Representation, Test, Consistency The Basic Terms: Representation  [.] 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:    V from PCP[D, V,  ) f is of global degree r

6. 5 Representation, Test, Consistency Test Test –A set of Boolean functions, –Each depends on at most D representation’s variables. local tests D from PCP[D, V,  )

7. 6 Representation, Test, Consistency 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.

8. 7 Geometry Let us define some specific affine subspaces of  : –lines(  ) is the set of all lines (affine subspaces of dimension 2) of  –planes(  ) is the set of all planes (affine subspaces of dimension 3) of 

9. 8 Overview of the Tests 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  The local-tests check compatibility between the values of different variables in  [.].

10. 9 Simple Test: Points-on-Line Representation Representation:  [.] has one variable  [p] for each point p . The variables are supposedly assigned the value ƒ(p). hence v = log |  |

11. 10 Points-on-Line: Test Test Test: There’s one local-test for each line l  lines(  ). Each 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 2r

12. 11 Points-on-Line: Consistency globally consistent Def: An assignment to  is said to be globally consistent if values on most points agree with a single, global degree-r polynomial. Thm[RuSu]: 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. Alas, each local-test depends on a non constant number of variables (2r)

13. 12 Next Test: Line-vs.-Point Representation Representation:  [.] 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

14. 13 Line-vs.-Point: Test Test Test: There’s one local-test for each pair of: –a line l  lines(  ), and –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.

15. 14 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.

16. 15 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?

17. 16 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)

18. 17 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 ].

19. 18 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.

20. 19 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.

21. 20 Limited Pluralism - Cont. 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.

22. 21 Plane-vs.-Plane: Representation Representation:  [.] has one variable  [p] for each plane p  planes(  ), supposedly assigned the restriction of f to p. Hence the range of  [p] is all degree-r two-variables poly’s

23. 22 Plane-vs.-Plane: Test Test: 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)log|  |. That is, a pair of plains intersecting by a line

24. 23 Plane-vs.-Plane: Consistency Thm[RaSa]: As long as   |  | -c for some constant 1 > c > 0, the tests err (w.r.t.  ) with a very small probability, namely   -c’ for some constant 1 > c’ > 0.

25. 24 Plane-vs.-Plane: Consistency - Cont. The theorem states that, the plane-vs.- plane test, with very high probability (  1 -  c’ ), either rejects, or accepts values of a  -permissible polynomial.

26. 25 Summary We examined consistency tests, Points-on-Line,Line-vs.-Point and Plane-vs.-Plane. By weakening to  -permissible definition, we achieve an error probability which is below constant.