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PCP Characterization of NP: Gap-QS[O(n), 2|α|-1] Sum Check Quadratic Equations

This text discusses the Gap-QS[O(n), 2|α|-1] problem, which involves distinguishing between satisfying and unsatisfying assignments for a set of quadratic equations. It explores the reduction and amplification claims, as well as the PCP Proof Map.

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PCP Characterization of NP: Gap-QS[O(n), 2|α|-1] Sum Check Quadratic Equations

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  1. PCP Characterization of NP: Proof chapter 3

  2. PCP Proof Map  Gap-QS[O(n),,2||-1] Sum Check quadratic equations of constant size with consistency assumptions Gap-QScons[O(1),,2||-1]   BUT it remains to prove the composition-recursion lemma... Consistent Reader conjunctions of constant number of quadratic equations, whose dependencies are constant. Gap-QS*[O(1),O(1),,||-] Error correcting codes Gap-QS[O(1),,2||-1]

  3. Today's road Using error correcting codes

  4. Conjunctions of Equations Definition (Gap-QS*[D1,D2,,]): Instance: a set of nconjunctions of D1 quadratic equations (polynomials) over . Each equation depends on at most D2 variables. Problem: to distinguish between: There is an assignment satisfying all the conjunctions. No more than an  fraction of the conjunctions can be satisfied simultaneously. YES NO

  5. Conjunctions of Equations An example instance of Gap-QS*[2,1,Z2,½]: Notice that we can satisfy more than a half of the equations!! Henceforth, we’ll assume the number of equations in all the conjunctions is the same. Is this a restriction?

  6. The reduction Claim: Gap-QS*[D1,D2,,] reduces to Gap-QS[D1·D2,,+||-1](as long as ||D1 is at most polynomial). Proof: Given an instance of Gap-QS*[D1,D2,,],replace each conjunction with all linear combinations of its polynomials.(i.e. apply Hadamard code to conjuncts)

  7. Correctness of the Reduction • If the original system had a common solution, so does the new system. • Otherwise, fix an assignment to the variables of the system and observe the two instances:

  8. polynomials originating from the blue set  polynomials originating from the pink set all satisfied  Analysis fraction of unsatisfied conjunctions fraction of satisfied polynomials originating from unsatifiable conjunctions  ||-1 fraction of satisfied conjunctions

  9. Relaxation • Yes instance of Gap-QS*[D1,D2,,] are transformed intoYes instances of Gap- QS[D1·D2,,+||-1]. • No instance of Gap-QS*[D1,D2,,] are transformed intoNo instances of Gap- QS[D1·D2,,+||-1]. • The construction is efficient when ||D1 is at most polynomial in the size of the input. • What proves the claim. 

  10. Amplification Claim:For any constant C, Gap-QS[D,,] reduces to Gap-QS[C·D,,C+||-1] (When || is at most polynomial in the size of the input). Proof: Given an instance of Gap-QS[D,,],generate the set of all linear combinations of C polynomials. By an argument similar to the former, the claim holds. 

  11. PCP Proof Map  Gap-QS[O(n),,2||-1] Sum Check quadratic equations of constant size with consistency assumptions Gap-QScons[O(1),,2||-1]   BUT it remains to prove the composition-recursion lemma... Consistent Reader conjunctions of constant number of quadratic equations, whose dependencies are constant. Gap-QS*[O(1),O(1),,||-]  Error correcting codes Gap-QS[O(1),,2||-1]

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