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For every integer k \geqslant 1, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability at most 2^{ - k^2+ 1}. In particular, the verifier achieves optimal amortized query complexity of 1+ \delta for arbitrarily small constant 娄\delta 0. Such a characterization was already proved by Samorodnitsky and Trevisan [15], but their verifier loses perfect completeness and their proof makes an essential use of this feature.By using an adaptive verifier we can decrease the number of query bits to 2k + k2, the same number obtained in [15]. Finally we extend some of the results to larger domains.

Query Efficient PCPs with Perfect Completeness