PCP Characterization of NP:

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PCP Characterization of NP Proof chapter 1
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PCP Proof Map
In previous lectures
3SAT
Clauses to polynomials
Solvability
Introducing new variables
QS
Error correcting codes
Gap-QSO(n),?,2?-1
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PCP Proof Map
In following lectures
Gap-QSO(n),?,2?-1
Sum Check
quadratic equations of constant size with
consistency assumptions
Gap-QSconsO(1),?,2?-1
Consistent Reader
conjunctions of constant number of quadratic
equations, whose dependencies are constant.
Gap-QSO(1),O(1),?,?-?
Error correcting codes
Gap-QSO(1),?,2?-1
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Today's road

• The sum check lemma

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Definitions

• Def Given a finite field ? and a positive
  parameter d, we define the corresponding domain
  as
• Fi xk k??d .
• The variables in the domain range over ?.

Def An assignment f?d?? to a domain is said to
be feasible if its a degree-r polynomial. Def
An assignment f?d?? to a domain is said to be
good if its a degree-s polynomial. (s?r and d
are some global constants.)
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Definitions

• Def (Gap-QSconsD,?,?)
• Instance A set of domains F1,...,Fk and n
  quadratic equations over ?. Each equation depends
  on at most D variables, some of them belong to
  some domains.
• Problem to distinguish between
• There is a good assignment satisfying all the
  equations.
• No more than an ? fraction of the equations
• can be satisfied simultaneously by a feasible
  assignment.

YES
NO
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Definitions
equation
x1 2 2 x2 x3 ... 3 xn 0
xn
x1
x3
promise

• Variables belong to some domains.
• The promise is that the values to a domains
  variables form a low-degree polynomial

(0,0,1,1)
(3,3,3,1)
(1,0,1,2)
domain
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The Sum-Check Lemma

• Lemma (Sum-Check)
• Gap-QSO(n),?,2/? is efficiently reducible to
  Gap-QSconsO(1),?,2/?.

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Overview

• We precede the proof by a general scheme
• Our starting point is the gap-QS instance, and we
  need to decrease (to constant) the number of
  variables each quadratic-polynomial depends on
• We will add variables to those of the original
  gap-QS instance, to check consistency, and
  replace each polynomial with many new ones
• The consistency will be checked later on in the
  proof utilizing the efficient consistent-readers
  we have seen
• Our test assumes the values for some preset sets
  of variables to correspond to the
  point-evaluation of a low-degree polynomial (an
  assumption to be removed by plugging in the
  consistent reader)

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Representing a Quadratic-Polynomial

• Given a quadratic-polynomial P, over variables
  Yi, let us write the value of P in a certain
  point in the space as follows

A is an assignment to the variables
( ?(i,j) is the coefficient of the monomial yiyj )
Let us convert the polynomial to linear form
lets assume a set of variables yij, i,j ?1..m,
with the intention that A(yij) A(yi) A(yj),
and the special case where A(yii) A(yi) which
lets us write
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Representing a Quadratic-Polynomial

• Next, we associate each variable yij with some
  point x?Hd. Define the following one-to-one
  function

Notice that
As a consequence we can define
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Representing a Quadratic-Polynomial

• Using the new definitions we can write

Where ? , A are functions
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Low Degree Extension (LDE)

• Def (low degree extension) Let ? Hd ? H be a
  string (where H is some finite field).
• Given a finite field F, which is a superset of H,
  we define a low degree extension of ? to F as a
  polynomial LDE? Fd ? F which satisfies
• LDE? agrees with ? on Hd (extension).
• The degree-bound of LDE? is H in each variable
  (low degree).

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Using the LDE
Let ƒ be a low-degree-extension of ? A
Notice that f is define by all ? A
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Using the LDE

• We therefore can write

Notice that LDE of both ? and A is of degree
H-1 in each variable, hence of total degree r
d(H-1), which makes ƒ of total degree 2r.
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Whats ahead

• We show next a test that uses a small number of
  variables
• For any assignment for which some variables
  corresponds to a function ƒ of degree 2r, the
  test verifies the sum of values of ƒ over Hd
  equals a given value.
• Each local-test accesses much smaller number than
  Hd of representation variables.
• Later on we will replace the assumption that ƒ is
  a low-degree-function by evaluating that single
  point accessed with an efficient
  consistent-reader for ƒ

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Partial Sums

• For any j?0..d define

That is, Sumƒ is the function that does not vary
on the first j variables, and sums over all
points for which the rest of the variables are
all in H PropositionSumƒ is of degree 2rd
Proof Immediate since ƒ is of degree 2r and
Sumƒ is the linear combination of d degree-r
functions
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Partial Sums

• PropositionFor every a1, .., ad ? ? and any
  j?0..d

Proof Homework...
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The Sum-Check Test

• Now we can assume Sumƒ to be of degree 2r (this
  is the consistency assumption to be verified
  later on with a consistent reader) and verify
  property 2, namely that for j0, Sumƒ gives the
  appropriate sum of values of ƒ
• RepresentationOne variable ?j , a1, .., ad
  for every a1, .., ad ? ? and j?0..d
  Supposedly assigned Sumƒ (j, a1, .., ad )(hence
  ranging over ? )
• TestOne local-test for every a1, .., ad ? ?
  one which accepts an assignment A if for every
  j?0..dA(?j,a1,..,ad) ?i?H
  A(?j1,a1,..,aj,i,aj2,..,ad)

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The Sum-Check Test

• The above test already drastically reduce the
  number of variables each linear-sum accesses to
  O(d H)
• However, we have introduces a consistency
  assumption (that the functions f are low degree)
• We shall now reduce the number of variables
  accessed to a constant O(1) (and strengthen the
  consistency assumption on the way) using the
  exact same method.

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The Sum-Check Test (repeated)

• Define the following function using the Sumƒ
  function defined previously, for a certain (a1,
  .., ad)

Using this notation, recall the sum-check test
was to verify that for a certain (a1, .., ad)
Define now a new function
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The Sum-Check Test (repeated)

• Claim If for a certain (a1, .., ad)

Then for a random uniform (i1, .., id)
Proof Homework
Note, from a function f with O(Hd) variables
weve received a function T with only O(rd)
variables.
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The Sum-Check Test (repeated)

• Claim If for a certain (a1, .., ad)

Then for a random uniform (i1, .., id)
Proof Homework
Using this fact we can now devise another local
test that relies on less variables