Consistency Tests

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 PCPD, V, ?)

• The Basic Terms
• Representation ?.
• ?. is a 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 representations
  variables.

D from PCPD, 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
• fs values on points of ?
• fs restriction to a line in ?
• fs 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 point p??.
• The variables are supposedly assigned the value
  ƒ(p)
• Hence the range of the variables is v
  log ?

9
Points-on-Line Test

• Test
• Theres one local-test for each line l?lines(?).
• 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

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
  consistent if values on most points agree with a
  single, global degree-r polynomial.
• ThmRuSu 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 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 polys
12
Line-vs.-Point Test

• Test
• Theres 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.

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-r polynomial.
• Here D O(1) V (r1) 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
Whats the problem?

• Adversary randomly partition variables into k
  sets, each consistent with a distinct degree-r
  polynomialThis would cause the local-tests
  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 .

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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 assignments values
  consistent with any ?-permissible ƒ are
  acceptable.

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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.

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Limited Pluralism - Cont.

• Note that the adversarys 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 plane
  p?planes(?),
• supposedly assigned the restriction of f to p.

Hence the range of ?p is all degree-r
two-variables polys
21
Plane-vs.-Plane Representation
22
Plane-vs.-Plane Test
That is, a pair of plains intersecting by a
line

• Test
• Theres 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 accepts if and only if the value of ?p1
  restricted to l equals the value of ?p2
  restricted to l.
• Here DO(1), v2(r1)2log?.

23
Plane-vs.-Plane Consistency

• ThmRaSaAs long as ? ³?-c for some constant
  1 gt c gt 0, a local test err (w.r.t. ?) with a
  very small probability, namely ?c for some
  constant 1 gt c gt 0.

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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 ?-permissible definition, we
  achieve an error probability which is below
  constant.