Revised: August 23, 2005

Published: September 28, 2005

**Keywords:**Computational complexity, approximation algorithms, probablistically checkable proofs, PCP, inapproximability, amortized query bits, perfect completeness

**Categories:**complexity theory, probabilistically checkable proofs, PCP, approximation algorithms, inapproximability, Fourier analysis

**ACM Classification:**F.2.2,

**AMS Classification:**68Q05

**Abstract:**
[Plain Text Version]

For every integer $k > 0$, and an arbitrarily small constant
$\epsilon>0$, we present a PCP characterization of NP where the
verifier uses logarithmic randomness, non-adaptively queries
$4k+k^2$ bits in the proof, accepts a correct proof with probability
1, i.e., it has perfect completeness, and accepts any supposed proof
of a false statement with probability at most $2^{-k^2}+\epsilon$.
In particular, the verifier achieves optimal *amortized query
complexity* of $1+\delta$ for arbitrarily small constant $\delta >
0$. Such a characterization was already proved by Samorodnitsky and
Trevisan (STOC 2000), 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 $2k+k^2$, equal to the number obtained by Samorodnitsky and Trevisan. Finally we extend some of the results to PCPs over non-Boolean alphabets.