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Consider verifiers in the usual sense:

Non adaptive : reads all positions of the proof predefined

Adaptive: Reads the proof iteratively.

Many lecture notes on PCP say the following: Adaptive verifier with q queries is equivalent to non-adaptive with some q' queries where q' is a constant and both using same randomness. In other words I think this means

Non adaptive q' queries can simulate adaptive q queries. But I don't see why? How to show this?

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Consider an adaptive algorithm making $q$ queries. Fixing the randomness, we can describe the positions queried by the algorithm using a function $A$ which gets as input the answers of the preceding queries, and outputs the next position to be queried. If the bits are read from an array $x$, then the adaptive algorithm works as follows:

  1. Read $b_1 = x_{A(\lambda)}$.
  2. Read $b_2 = x_{A(b_1)}$.
  3. Read $b_3 = x_{A(b_1b_2)}$.
  4. ...
  5. Read $b_q = x_{A(b_1 b_2 \ldots b_{q-1})}$.

To simulate this non-adaptively, we simply read $x_{A(w)}$ for all words $w$ of length at most $q-1$. Given these $q' = 2^q-1$ non-adaptive queries, we can simulate the adaptive algorithm.

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