Verifying Membership in NP-languages, or How to Avoid Reading Long Proofs

The class NP was de ned by Cook ([Coo71]) and independently by Levin ([Lev73]) as the class of languages that are accepted in non-deterministic polynomial time. If L 2 NP and x 2 L, then there is a \proof" that x 2 L which can be veri ed in deterministic polynomial time (the \proof" consists merely of the choices made by the non-deterministic Turing machine). For example, for the problem of satis ability of a Boolean formula, the \proof" can be a satisfying truth-assignment. Then it can be veri ed by simply plugging the truth-values into the formula. The class IP is de ned as the class of languages which have e cient interactive proofs of membership, and was introduced in [GMR89] and in [Bab85]. In the IP model for a language L, given input x, a veri er V is communicating with a prover P who is trying to convince V that x 2 L. The Prover can have unlimited computational powers, whereas the Veri er is restricted to randomized polynomial-time computations. L 2IP if for all x 2 L, the Veri er can be convinced that x 2 L; and if x = 2 L, then no Prover P 0 can convince V otherwise with a non-negligible probability. Building on the ideas of Lund, Fortnow, Karlo and Nisan ([LFKN92]), Shamir, in [Sha90], proved that IP=PSPACE. That is, the languages for which the membership can be established in polynomial space (and any amount of time) are the same as the languages for which membership can be interactively proved in polynomial time. The IP model lead to the de nition of the class MIP (for multi-prover interactive proofs) in [BGKW88]. In the MIP model, two or more provers that cannot communicate with each other interact with the Veri er. In [FRS88], MIP was proved equivalent to the following model. LetM be a random polynomial-time Turing machine that can interact with an oracle Ox. The oracle is trying to convince M that x 2 L. An oracle, unlike a prover, is memoryless,