Consistency Tests

1
Consistency Tests

• for Low-degree Polynomials

2
ConsistencyRepresentation, Local, Global

• Representation a set of variables ?.whose
  assigned values (in the range 2v) all supposedly
  correspond to a single, global degree-r
  polynomial ƒ ? a ?
• Test a set of Boolean functions, each depending
  on at most D representations variables (hence
  referred to as local-tests)
• Consistency the fraction of local-tests
  satisfied being at least ? implies the assigned
  values conform to some notion of global
  consistency (to be precisely specified)

3
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 ?
• we are now ready to describe the first test

4
Simple Test Points-on-Line

• Representationone variable ?p for every point
  p ??which is supposedly assigned ƒ(p) hence v
  log ?
• Testone local-test for each line l ?
  lines(?)which depends on all (or 2r) points on
  l, andaccepts if and only if values consistent
  with a single degree-r univariate polynomial

5
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
• ThmRuSu If large (constant) fraction of
  local-test accept, then there is a degree-r ƒ
  which agrees with the assigned values on most
  points Proof omitted. (Not hard.) (Neither
  trivial.)
• Alas, each local-test depends on non constant
  number of variables

6
Next Test Line-vs.-Point

• Representation
• 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 ranging hence over all degree-r univariate
  polys
• Test
• one local-test for each pair ofa line l ?
  lines(?) and a point p ? l
• which accepts if value assigned to ?p equals
  the value of the polynomial assigned to ?l on p

7
Global Consistency Constant Error

• Thm AS,ALMSS Probability of finding
  inconsistency, between value for ?p and value
  for line ?l on p, high (constant) unlessmost
  lines and most points agree with a single,
  global degree-r polynomial.
• Here D O(1) ! V (r1) log? ? constant

8

• Prove that if restriction of ƒ ? a ? to every
  line of ? is a univariate polynomial of degree r,
  then ƒ itself is of degree r
• Bonus if almost all lines are of degree r show ƒ
  agrees with a degree-r g on almost all points of ?

9
Can the test be improved?

• Can error-probability be made smaller than
  constant, while keeping each local-test depending
  on constant number of representation variables?
• 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 if all
  variables fall within the same set in partition

10
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

11
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. Formally
• Def A local test is said to err (with respect to
  ?) if it accepts values that are inconsistent
  with all ?-permissible degree-r ƒ s

12
Plane-vs.-Plane Test

• Representation
• one variable for each plane p ?
  planes(?)supposedly assigned the restriction of
  ƒ to p ranging hence over all dimension-2,
  degree-r polynomials
• Test
• One test for every line l ? lines(?) and pair of
  planes p1, p2 ? planes(?) such that l ? p1 and l
  ? p2 i.e., a pair of planes that intersect by a
  line
• Accept if and only if value for ?p1 restricted
  to l equals value of ?p2 restricted to l
• Here DO(1), v2(r1)log?. What is ? ?

13
Plane-vs.-Plane Test

• ThmRaSaas long as ? ³?-c for some constant
  1 gt c gt 0test errs (w.r.t. ?) with very small
  probability, namely ?-c for some constant 1 gt
  c gt 0
• 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
• ShowThe number of ?-permissibledegree-r
  polynomials is at most ?-2.