Consistent Readers. Read Consistently a value for arbitrary points. Introduction. We are going to use several consistency tests for Consistent Readers. Plane Vs. Point Test - Representation. Representation :
Read Consistently a value for arbitrary points
planepand a pointx on p.
Reminder:A: planes dimension-2 degree-r polynomial
The error probability of this test is very small, i.e.< c’/2 , for some known 0<c’<1.
The error probability is the fraction*of pairs <x, p> for a
point x and plane p whose:
* fraction from the set of all combination of (point, plane)
By reduction to Plane-Vs.-Planetest:
Let’s denote this test by PPx-Test
What is its error-probability?
Proposition: The error-probability of PPx-Test is “almost the same“as Plane-Vs.-Plane’s.
The test errs in one of two cases:
So Pr(Second-Case Error) £ r/||
PPx-Test’s error-probability £ c + r/||
For an appropriate (namely: cO(r/||)):
c + r/|| = O(c)
So, PPx-Test’s error-probability is
£ c’, for some 0<c’<1
Back to Plane-Vs.-Point:
Plane-Vs.-Point’s error probability is:
Pr p, x ((A(p))(x) = A(x) ) =
= Pr lx, p1 ( (A(p1))(x) = A(x) )
Prp, x ((A(p))(x) = A(x) )
= Prlx, P1 ( (A(p1))(x) = A(x) )
=* Elx ( Prp1 ( (A(p1))(x) = A(x)|xl ) )
=**Elx( (Prp1, p2 ( (A(p1))(x) = (A(p2))(x) = A(x) | xl ) )1/2 )
( Elx(Prp1, p2 ((A(p1))(x) = (A(p2))(x) = A(x) | xl ) )1/2
*( Prlx, p1, p2 ((A(p1))(x) = (A(p2))(x) = A(x))1/2
*event A, and random variable Y, Pr(A) = EY( Pr(A|Y) )
** Prp1, p2 ( (A(p1))(x) = (A(p2))(x) = A(x) | xL ) ) = (p1,p2 are independent)
(Prp1 ( (A(p1))(x) = A(x) | xl ) )* (Prp1 ( (A(p2))(x) = A(x) | xl ) ) =
(Prp1 ( (A(p1))(x) = A(x) | xl ) )2
We’ve established that:
Plane-Vs.-Point error probability, i.e.,
The probability that p(which israndom)is
is < c’/2.
Note: This proof is only valid as long as the point x whose value we would like to read is random.
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(<c’)
Such procedure is called:consistent-reader.
A local reader, can either reject or return a value
from the field .
[supposedly the value is ƒ(x), with ƒ a degree-r polynomial].
Representation: One variable for each plane.
Claim:Withhighprobability ( 1-c’) R [x]either rejects or returns a permissible value for x.
[i.e., consistent with one of the permissible polynomials].
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 differentpermissible polynomials.
Produces a distribution very close to uniform over planes pAlso, p w.h.p. does not contain a point of .
Error Probability: c’/2
The error probability doesn’t increase.
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.