PCP Characterization of NP:

1
PCP Characterization of NP Proof chapter 2
2
PCP Proof Map
?
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
3
In this lecture

• Using consistent readers

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Consistent Readers Remove Consistency Assumptions

• Concept Readers are plugged into equations, and
  ensure (w.h.p) the domain variables are only
  assigned values corresponding to some
  ?-permissible polynomial.
• Def A function f is called a ?-permissible
  polynomial (with respect to an assignment A) if
  at least 1- ? of the points x are assigned the
  value of a dimension d and degree r polynomial.

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Consistent Readers Scheme
Q How do the consistency gadgets weve seen fit
into this scheme?
Consistent Reader
Local Readers
Representation
Local Reader
Local Reader
...
...
variable
variable
Local Test
Evaluator
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The Desired Properties of A Consistent Reader

• Efficiency There is only a polynomial number of
  local readers.
• Low Dependency Each local reader depends on a
  constant number of representation variables.
• Linearity Each local test is a conjunction of
  linear equations over representation variables.
  Each evaluator is a linear combination of such.
• Range All variables should range over ?.

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The Desired Properties of A Consistent Reader

• Completeness All local tests accept every
  assignment to the variables corresponding to some
  ?-permissible polynomial. Moreover, every
  evaluator must return the evaluations of that
  polynomial.
• Soundness At most an ? fraction of the tests can
  be satisfied when no ?-permissible polynomial
  exists.

8
The Consistent Reader We Have

• Representation
• One variable for every cube (affine subspace) of
  dimension k2, containing the k points
  corresponding to variables from this domain
  appearing in equations.(Values of the variables
  range over all degree-r, dimension k2
  polynomials)
• One variable for every point x??d.(Values of the
  variables range over ?).

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The Consistent Reader We Have

• Consistent-Reader
• One local-reader for every cube-variable C and a
  point-variable 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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Does This Consistent Reader Suffice?

• Implicitly, our consistent reader relies itself
  on consistency assumptions.
• This will be resolved in the proof of the
  Composition-Recursion lemma, assuring us there
  exists a satisfactory consistent reader.
• In the meantime, let us assume that, and continue
  with the PCP proof.

Youd might want to take a better look at the
desired properties and the construction first.
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Gap-QScons Reduces To Gap-QS
Note the uniformity of the construction Each
equation contributes the exact same number of
conjunctions.

• The Construction
• Let p be one of the equations.
• Assume there are k domains F1,...,Fk (associated
  with k consistent readers).
• Generate a conjunction for every choice of one
  local reader of each consistent reader.
• Each conjunction has the following format

the original equation, where each domain variable
is replaced by its evaluation
?
?
?
local test 1
local test k
...
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Correctness Completeness

• Let A denote an instance ofGap-QSconsO(1),?,2/?
  .
• Let B denote the corresponding instance of
  Gap-QSO(1),O(1),?,?-?.
• Completeness a good satisfying assignment for A
  can be easily extended into a satisfying
  assignment for B.

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How Are We Going To Prove Soundness?
Well show that for any assignment, more than few
satisfied conjunctions imply more than few
satisfied equations even under the consistency
assumption.
conjunctions
contributes
equations
satisfied
were only interested in equations, for which
this fraction is big.
Such can be satisfied when ?-permissible
polynomials are assigned to their domains.
Since therere few permissible polynomials, even
a random choice should satisfy enough equations.
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Correctness Soundness

• Suppose there exists an assignment which
  satisfies ?gt?-? of Bs conjunctions.
• Let p be an equation which more than k? of its
  conjunctions are satisfiable.
• At least ?-k? of the equations satisfy this.

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Correctness Soundness

• For every reader, at most ? of the local tests
  err.
• Overall at most k? of all local tests err.
• Hence there exists a satisfied conjunction with
  no erroneous tests.
• ? ? satisfying assignment for p which assigns
  every domain a ?-permissible polynomial.

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Auxiliary Lemma Not Many Permissible Polynomials

• Lemma For any big enough ? (?2gt4rd?-1), there
  are less than 2?-1 ?-permissible polynomials.
• Proof
• Suppose there were 2?-1 ?-permissible
  polynomials.
• Each agrees with every other polynomial on at
  most rd?-1 of the points.

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Auxiliary Lemma Not Many Permissible Polynomials

• Overall on at most a 2?-1rd?-1 fraction there
  exists some polynomial it agrees with.
• Which is less than ½? by the choice of ?.
• But a ?-permissible polynomial agrees with the
  assignment on at least ? of the points.

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Auxiliary Lemma Not Many Permissible Polynomials

• Thus, for each polynomial for more than ½? of the
  points its the only one agreeing with the
  assignment on them.
• Contradiction! Recall that by our assumption
  there are as many as 2?-1 polynomials. ?

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Proof of Soundness Continued

• If we assign each domain a random ?-permissible
  polynomial, p should be satisfied with
  probability at least O(?k).
• Hence the expected number of satisfied equations
  is at least O((?-k?)?k).
• Thus at least one assignment satisfies that many
  equations.
• For an appropriate choice of the parameter ?,
  this number is greater than 2/?. ?

20
PCP Proof Map
?
Gap-QSO(n),?,2?-1
Sum Check
quadratic equations of constant size with
consistency assumptions
Gap-QSconsO(1),?,2?-1
?
? BUT it remains to prove the composition-recursio
n lemma...
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