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Non-Deterministic Exponential Time has Two-Prover Interactive Protocols
| Citations: | 413 - 37 self |
BibTeX
@MISC{Babai_non-deterministicexponential,
author = {Laszlo Babai and Lance Fortnow and Carsten Lund},
title = { Non-Deterministic Exponential Time has Two-Prover Interactive Protocols},
year = {}
}
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Abstract
We determine the exact power of two-prover inter-active proof systems introduced by Ben-Or, Goldwasser, Kilian, and Wigderson (1988). In this system, two all-powerful non-communicating 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 two-prover interactive proof systems is nondeterministic exponential time. After the recent results that all languages in PSPACE 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 NEXP # NP. In particular, for the first time, prov-ably polynomial time intractable languages turn out to admit “efficient proof systems’’ since NEXP # P. We show that to prove membership in languages in EXP, the honest provers need the power of EXP only. A consequence, linking more standard concepts of structural complexity, states that if EX P has poly-nomial size circuits then EXP = Cg = MA. The first part of the proof of the main result ex-tends 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 n-variable 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.
Keyphrases
interactive protocol non-deterministic exponential time second part standard concept conp-complete language all-powerful non-communicating provers nexp np polynomial time exp cg ma verification scheme first part polynomial time verifier structural complexity input belongs n-variable function single prover interactive proof program verification truth value efficient proof system immense power efficient provability two-prover interactive proof system sharp contrast relativized sense poly-nomial size circuit polynomial extrapolation nondeterministic exponential time single prover case recent result combinatorial technique proof rest multiple provers two-prover inter-active proof system independent result proof system main result ex-tends recent technique first time expansion rate exact power polynomial time intractable language
