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I'm looking through some lectures and books to understand the PCP verifier constructed by Hastad. I noticed that the subtle difference of an adaptive or non-adaptive PCP verifier seems to correspond to verifier with perfect or non-perfect completeness.

In "A tight characterization of NP with 3 query PCPs" the authors point out that there are (due to Trevisan and Zwick) strong restrictions using non-adaptive PCPs, e.g. $$P = PCP_{1, \frac{5}{8} + \epsilon}(log(n), 3)$$ and they construct an adaptive PCP finally showing that $NP = PCP_{1, \frac{1}{2}+\epsilon}(log (n), 3)$.

Then again the definition in the textbook from Arora, Barak (and in other lectures) uses explicitly non-adaptive verifiers with perfect completeness (well, with soundness $\frac{1}{2}$).

Could someone give further explanations on this? Are different definitions with soundness paramter $\frac{1}{2}$ equivalent?

And one further question: Is it possible to construct a linear dictatorship test (as basis of the Hastad Verifier) which is complete AND non-adaptive?

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  • $\begingroup$ Does the Arora-Barak definition you refer to use 3 queries, or a larger constant? $\endgroup$
    – Or Meir
    Nov 1, 2014 at 21:10
  • $\begingroup$ The Hastad verifier in the book makes exactly 3 queries, is non-adaptive and has no full completeness. But with "definition" I meant the general definition of PCP verifiers, not specifically the Hastad verifier. $\endgroup$ Nov 1, 2014 at 21:17

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