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Non-deterministic exponential time has two-prover interactive protocols. (English) Zbl 0774.68041
Summary: We determine the exact power of two-prover interactive proof systems introduced by M. Ben-Or, S. Goldwasser, J. Killian and A. Wigderson [Multi-prover interactive proofs: How to remove the intractability assumptions, in Proc. 20th Ann. ACM Symp. Theory of Computing, 113-131 (1988)]. In this system, two all-powerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input $$x$$ belongs to the language $$L$$. We show that the class of languages having two-prover interactive proof systems in nondeterministic exponential time.
We also show that to prove membership in languages in $$EXP$$, the honest provers need the power of $$EXP$$ only.
The first part of the proof of the main result extends recent techniques of polynomial extrapolation used in the single prover case by C. Lund, L. Fortnow, H.Karloff and N.Nisan [Algebraic methods for interactive proof systems, in Proc. 31st Ann. IEEE Symp. Foundations of Comp. Sci., 1-10 (1990)] and A. Sharmir [IP=PSPACE, in Proc. 31st Ann. IEEE Symp. Foundations of Comp. Sci., 11-15 (1990)]. The second part is a verification scheme for multilinearity of the function in several variables held by an oracle and can be viewed as an independent result on program verification. Its proof rests on combinatorical techniques employing a simple isoperimetric inequality for certain graphs.

##### MSC:
 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q60 Specification and verification (program logics, model checking, etc.)
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