Consistency Tests for Low Degree Polynomials

1. Consistency Tests for low degree polynomials

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. We present the tests: • Points-on-Line • Line-vs.-Point • Plane-vs.-Plane

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

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

5. Basic Terms • 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. Affine subspaces • 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 • 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 [.].

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

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

10. Points-on-Line: Consistency Alas, each local-test depends on a non constant number of variables (2r) Def: An assignment to is said to be globally consistentif 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.

11. Next Test: Line-vs.-Point Representation: • [.] has one variable [p] for each pointp, supposedly assignedƒ(p), • Plus, one variable [l] for each linellines(),supposedly assignedƒ’srestriction tol. Hence the range of [l] is all degree-r univariate poly’s

12. Line-vs.-Point: Test Test: • There’sone local-test for each pair of: • a line l  lines(), and • a point p  l . • A test acceptsif 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) , unlessmost lines and most points agree with a single, global degree-rpolynomial. HereD = 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 polynomialThis 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 deducePCPcharacterization ofNP].

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-rƒ’s.

19. 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.

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

21. Plane-vs.-Plane: Representation

22. That is, a pair of plains intersecting by a line Plane-vs.-Plane: Test Test: • There’s one local-test for each line llines() and a pair of planes p1,p2planes() such that lp1 and lp2 • A test acceptsif and only if the value of[p1]restricted tol equals the value of[p2]restricted to l. HereD=O(1), v=2(r+1)2log||.

23. Plane-vs.-Plane: Consistency Thm[RaSa]: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 • We examined consistency tests, Points-on-Line,Line-vs.-Point and Plane-vs.-Plane. • By weakening to-permissibledefinition, we achieve an error probability which is below constant.