2. 1

3. 2 Introduction In this chapter we examine consistency tests, and trying to improve their parameters: In this chapter we examine consistency tests, and trying to improve their parameters: reducing the number of variables accessed by the test. reducing the number of variables accessed by the test. reducing the variables’ range. reducing the variables’ range. reducing error probability. reducing error probability. We present the tests: Points-on-Line Points-on-Line Line-vs.-Point Line-vs.-Point Plane-vs.-Plane Plane-vs.-Plane

4. 3 Basic Terms The Basic Terms: Representation  [.] Representation  [.]  [.] is a set of variables, for which a value is assigned,  [.] is a set of variables, for which a value is assigned, The values are in the range 2 v, The values are in the range 2 v, The values correspond to a single, polynomial ƒ:    of global degree r The values correspond to a single, polynomial ƒ:    of global degree r V from PCP[D, V,  )

5. 4 Basic Terms Test Test A set of Boolean functions (local tests) A set of Boolean functions (local tests) Each depends on at most D representation’s variables. Each depends on at most D representation’s variables. D from PCP[D, V,  )

6. 5 Basic Terms Consistency: Consistency: Measures an amount of conformation between the different values assigned to the representation variables. 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. We say that the values are consistent if they satisfy at least an  -fraction of the local tests.

7. 6 Affine subspaces Let us define some specific affine subspaces of  : Let us define some specific affine subspaces of  : lines(  ) is the set of all lines (affine subspaces of dimension 1) 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  planes(  ) is the set of all planes (affine subspaces of dimension 2) of 

8. 7 Overview of the Tests In each tests the variables in  [.] represent some aspect of the given polynomial f, such as In each tests the variables in  [.] represent some aspect of the given polynomial f, such as f’s values on points of  f’s values on points of  f’s restriction to a line in  f’s restriction to a line in  f’s restriction to a plane in  f’s restriction to a plane in  The local-tests check compatibility between the values of different variables in  [.]. The local-tests check compatibility between the values of different variables in  [.].

9. 8 Simple Test: Points-on-Line Representation:  [.] has one variable  [p] for each point p .  [.] has one variable  [p] for each point p . The variables are supposedly assigned the value ƒ(p) The variables are supposedly assigned the value ƒ(p) Hence the range of the variables is: Hence the range of the variables is: v = log |  |

10. 9 Points-on-Line: Test Test: There’s one local-test for each line l  lines(  ). There’s one local-test for each line l  lines(  ). Each test depends on all points of l (altogether 2r points). Each test depends on all points of l (altogether 2r points). A test accepts if and only if the values are consistent with a single degree-r univariate polynomial A test accepts if and only if the values are consistent with a single degree-r univariate polynomial

11. 10 Points-on-Line: Consistency 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)

12. 11 Next Test: Line-vs.-Point Representation:  [.] has one variable  [p] for each point p , supposedly assigned ƒ(p),  [.] 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. 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

13. 12 Line-vs.-Point: Test Test: There’s one local-test for each pair of: There’s one local-test for each pair of: a line l  lines(  ), and a line l  lines(  ), and a point p  l. 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. A test accepts if the value assigned to  [p] equals the value of the polynomial assigned to  [l] on the point p.

14. 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.

15. 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?

16. 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)

17. 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 ].

18. 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.

19. 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.

20. 19 Limited Pluralism - Cont. Note that the adversary’s randomly partition does not trick the test this time: 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. If the test accepts when all the variables are from a set consistent with an r-degree polynomial, then the polynomial is really  - permissible.

21. 20 Plane-vs.-Plane: Representation Representation:  [.] has one variable  [p] for each plane p  planes(  ),  [.] has one variable  [p] for each plane p  planes(  ), supposedly assigned the restriction of f to p. supposedly assigned the restriction of f to p. Hence the range of  [p] is all degree-r two-variables poly’s

22. 21 Plane-vs.-Plane: Representation

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 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. 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|  |. 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, a local test 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. 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. By weakening to  -permissible definition, we achieve an error probability which is below constant.