We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that ... more >>>

It is known that there exists a PCP characterization of NP where the verifier makes 3 queries and has a {\em one-sided} error that is bounded away from 1; and also that 2 queries do not suffice for such a characterization. Thus PCPs with 3 queries possess non-trivial verification power ... more >>>

The error probability of Probabilistically Checkable Proof (PCP) systems can be made exponentially small in the number of queries by using sequential repetition. In this paper we are interested in determining the precise rate at which the error goes down in an optimal protocol, and we make substantial progress toward ... more >>>