Non-deterministic exponential time has two-prover interactive protocols

L�szl� Babai, Lance Fortnow, Carsten Lund
1991 Computational Complexity  
We determine the exact power of two-prover interactive proof systems introduced by Ben-Or, Goldwasser, Kilian, and Wigderson (1988) . In this system, two all-powerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language L. It was previously suspected (and proved in a relativized sense) that coNP-complete languages do not admit such proof systems. In sharp contrast, we show that the class of languages having
more » ... r interactive proof systems is nondeterministic exponential time. After the recent results that all languages in P S P A C E have single prover interactive proofs (Lund, Fortnow, Karloff, Nisan, and Shamir), this represents a further step demonstrating the unexpectedly immense power of randomization and interaction in efficient provability. Indeed, it follows that multiple provers with coins are strictly stronger than without, since N E X P # N P . In particular, for the first time, provably polynomial time intractable languages turn out to admit "efficient proof systems" since N E X P # P. We show that t o prove membership in languages in EXP, the honest provers need the power of E X P only. A consequence, linking more standard concepts of structural complexity, states that if E X P has polynomial size circuits then E X P = Cg = MA. The first part of the proof of the main result extends recent techniques of polynomial extrapolation of truth values used in the single prover case. The second part is a verification scheme for multilinearity of an nvariable function held by an oracle and can be viewed as an independent result on program verification. Its proof rests on combinatorial techniques including the estimation of the expansion rate of a graph. ~ Theorem 1.1 M I P = N E X P . In other words, the set of languages with two-prover interactive proof syst e m s is exactly the set of languages computable in nondeterministic exponential time. BGKW [7] in fact shows that all languages that
doi:10.1007/bf01200056 fatcat:7fo5g7yc5jdr3mpfeyh5bsb6ey