Consistent Readers

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Consistent Readers

Read Consistently a value for arbitrary points
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Introduction

• We are going to use several consistency tests for
  Consistent Readers.

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Plane Vs. Point Test - Representation

• Representation
• One variable for each plane p of planes(?),
  supposedly assigned the restriction of to p.
  (Values of the variables rang over all
  2-dimensional, degree-r polynomials).
• One variable for each point x ? ?. (Values of the
  variables rang over the field ?).

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Plane Vs. Point Test - Test

• Test
• One local-test for every
• plane p and a point x on p.
• Accept if
• As value on x, and
• As value on p restricted to x are consistent.
• ReminderA planes ? dimension-2 degree-r
  polynomial

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Plane Vs. Point Test Error Probability

• Claim
• The error probability of this test is very small,
  i.e. lt ?c/2 , for some known 0ltclt1.
• The error probability is the fraction of pairs
  ltx, pgt for a
• point x and plane p whose
• As value are consistent, and yet
• Do not agree with any ?-permissible degree-r
  polynomial (on the planes),
• fraction from the set of all combination of
  (point, plane)

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Plane Vs. Point Test Error Probability - Proof

• Proof
• By reduction to Plane-Vs.-Plane test
• replace every
• Local-test for p1 p2 that intersect by a line
  l,
• by a
• Set of local-tests, one for each point x on l,
  that compares p1s p2s values on x.
• Lets denote this test by PPx-Test
• What is its error-probability?

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Plane Vs. Point Test Error Probability - Proof
Cont.

• Proposition The error-probability of PPx-Test is
  almost the same as Plane-Vs.-Planes.
• Proof
• The test errs in one of two cases
• First case
• p1 p2 agree on l, but
• Have impermissible values (i.e. they do not
  represent restrictions of 2 ?-permissible
  polynomials).
• Second case
• p1 p2 do not agree on l, but
• Agree on the (randomly) chosen point x on l.

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Plane Vs. Point Test Error Probability - Proof
Cont.

• In the first case Plane-Vs.-Plane also errs, so
  according to RaSa, for some constant 0ltclt1
  Pr(First-Case Error) ?c
• For the second case, recall that
• r points, that two r-degree, 1-dimensional
  polynomials can agree on.
• ? points on the line l.
• So Pr(Second-Case Error) r/?
• ?PPx-Tests error-probability ?c r/?

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Plane Vs. Point Test Error Probability - Proof
Cont.

• For an appropriate ? (namely ?c?O(r/?))
• ?c r/? O(?c)
• So, PPx-Tests error-probability is
• ?c, for some 0ltclt1

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Plane Vs. Point Test Error Probability - Proof
Cont.

• Back to Plane-Vs.-Point
• Let p?planes, x?(points on p), such that
• A(p) and A(x) are impermissible.
• Let l?lines such that x ? l
• Let p1, p2 be planes through l
• Plane-Vs.-Points error probability is
• Pr p, x ( (A(p))(x) A(x) )
• Pr l?x, p1 ( (A(p1))(x) A(x) )

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Plane Vs. Point Test Error Probability - Proof
Cont.

• Prp, x ( (A(p))(x) A(x) )
• Prl?x, P1 ( (A(p1))(x) A(x) )
• El?x ( Prp1 ( (A(p1))(x) A(x) x?l ) )
• El?x ( (Prp1, p2 ( (A(p1))(x) (A(p2))(x)
  A(x) x?l ) )1/2 )
• ? ( El?x (Prp1, p2 ( (A(p1))(x) (A(p2))(x)
  A(x) x?l ) )1/2
• ? ( Prl?x, p1, p2 ( (A(p1))(x) (A(p2))(x)
  A(x) )1/2
• ? (?c)1/2
• ?event A, and random variable Y, Pr(A) EY(
  Pr(AY) )
• Prp1, p2 ( (A(p1))(x) (A(p2))(x) A(x)
  x?L ) ) (p1,p2 are independent)
• (Prp1 ( (A(p1))(x) A(x) x?l ) ) (Prp1 (
  (A(p2))(x) A(x) x?l ) )
• (Prp1 ( (A(p1))(x) A(x) x?l ) )2
• PPx-Test

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Plane Vs. Point Test Error Probability - Proof
Cont.

• Conclusion
• Weve established that
• Plane-Vs.-Point error probability, i.e.,
• The probability that p (which is random) is
• Assigned an impermissible value, and
• This value agrees with the value assigned to x
  (which is also random),
• is lt ?c/2.
• Note This proof is only valid as long as the
  point x whose value we would like to read is
  random.

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Reading an Arbitrary Point

• Can we have similar procedure that
• would work for any arbitrary point x?
• i.e., a set of evaluating functions, where the
  function
• returns an impermissible value with only a small
  (lt?c)
• probability.
• Such procedure is called consistent-reader.

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Consistent Reader for Arbitrary Point

• Representation As in Plane-Vs-Point test.
• local-readers Instead of local-tests, we have a
  set of (non Boolean) functions, ?x
  ?1,...,?m, referred to as local-readers.
• A local reader, can either reject or return a
  value
• from the field ?.
• supposedly the value is (x), with a degree-r
  polynomial.

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3-Planes Consistent Reader for a Point x

• Representation One variable for each plane.
• Consistent-Reader
• For a point x, ?x has one local-reader ?p2,
  p3 for every pair of planes p2 p3 that
  intersects by a line l.
• Let p1 be the plane spanned by x and l, ?p2, p3
• rejects, unless As values on p1, p2 p3 agree
  on l,
• otherwise returns As value on p1 restricted to
  x.

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Consistency Claim

• Claim With high probability ( ? 1-?c)? ?R
  ?x either rejects or returns a permissible
  value for x.
• i.e., consistent with one of the permissible
  polynomials.
• Remarks
• The sign ?R is used for randomly select from.
• Note that randomly selecting X and using it with
  l to span P1 is equal to randomly selecting l in
  P1 .

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Consistency Proof
with high probability

• Proof
• The value A assigns l, according to p2 p3s
  values, is permissible w.h.p. (1-?c).
• On the other hand, l is a random line in p1 and
  if p1 is assigned an impermissible value (by A),
  then that value restricted to most ls would be
  impermissible.

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Consistent-Reader for Arbitrary k points

• How can we read consistently more than one value
  ?
• Note Using the point-consistent-reader, we need
  to invoke the reader several times, and the
  received values may correspond to different
  permissible polynomials.
• Let ? x1, .., xk be tuple of k point of the
  domain ?,
• ? ? ?1, .., ?m is now set of functions,
  which can either reject or evaluate an assignment
  to x1, .., xk.

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Hyper-Cube-Vs.-Point Consistent-Reader For k
Points

• Representation
• One variable for every cube (affine subspace) of
  dimension k2, containing ?.(Values of the
  variables rang over all degree-r, dimension k2
  polynomials )
• one variable for every point x ??.(Values of the
  variables rang over ? ).

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Hyper-Cube-Vs.-Point Consistent-Reader For k
Points

• Show that the following distribution
• Choose a random cube C of dimension k2
  containing ?
• Choose a random plane p in C
• Return p
• Produces a distribution very close to uniform
  over planes pAlso, p w.h.p. does not contain a
  point of ?.

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Consistent Reader For k Values - Cont.

• Consistent-Reader
• One local-reader for every cube C containing ?
  and a point y ? C, which
• rejects if As value for C restricted to y
  disagrees with As value on y,
• otherwise returns As values on C restricted to
  x1, .., xk.

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Proof of Consistency

• Error Probability ?c/2
• Suppose,
• We have, in addition, a variable for each plane,
• The test compares As value on the cube C
• against As value on a plane p, and then
• against a point x on that plane.
• The error probability doesnt increase.

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Proof of Consistency - Cont.

• Proposition This test induces a distribution
  over the planes p which is almost uniform.
• Lemma Plane-Vs.-Point test works the same if
  instead of assigning a single value, one assigns
  each plane with a distribution over values.

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Summary

• We saw some consistent readers and how accurate
  they are. They will be a useful tool in this
  proof.