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ACM Symposium on Theory of Computing (STOC)

Two-prover one-round proof systems

Uriel Feige, László Lovász

Published Jan 1, 1992• 161 citations
Publisher

Abstract

We characterize the power of two-prover one-round (MIP(2,1)) proof systems, showing that MIP(2,1)=NEXPTIME. However, the following intriguing question remains open: Does parallel repetition decrease the error probability of MIP(2,1) proof systems?.
DOI
10.1145/129712.129783

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We show that no fixed number of parallel repetitions suffices in order to reduce the error in two-prover one-round proof systems from one constant to another. Our results imply that the recent bounds proven by Ran Raz, showing that the number of rounds that suffice is inversely proportional to the answer length, are nearly best possible. Our proof technique builds upon an idea of Oleg Verbitsky. We use this opportunity to survey the known results on parallel repetition, and to present the proofs of some previously claimed theorems. 1 Introduction A two prover one round proof system [8], MIP(2,1), is a protocol by which two provers jointly try to convince a computationally limited probabilistic verifier that a common input belongs to a prespecified language. The verifier selects a pair of questions at random. Each prover sees only one of the two questions, and sends back an answer. The verifier evaluates a predicate on the common input and the two questions and answers, and accepts or ...
We show that a parallel repetition of any two-prover one-round proof system (MIP(2,1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible [U. Feige and O. Verbitsky, Proc.11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 70--76].
We show that a parallel repetition of two-prover oneround proof systems (MIP(2, 1)) reduces the probability of error at an exponential rate.This settles a standing open problem, sometimes called 'The Parallel Repetition Conjecture".Previously, no constructive bound was known.The constant we have in the exponent is logarithmic in the total number of possible answers of the two provers.This is significantly better than what was previously known even in the much stmpler case where the questions to the two provers are chosen independently at random.In the most frequently used cases, our constant is just a global constant.
We introduce the miss-match form for two-prover one-round proof systems.Any two-prover one-round proof system can be easily modified so as to be in miss-match form.Proof systems in miss-match form have the projection property that is important for deriving hardness of approximation results for NP-hard combinatorial optimization problems. Our main result is an upper bound on the number of parallel repetitions that suffice in order to reduce the error of miss-match proof systems from p to � .This upper bound depends only on p and on � (polynomial in 1/(1 − p) and in 1/� ).Based on previous work, it follows that for any �> 0, NP has two-prover one-round proof systems with logarithmic-sized questions, constant-sized answers, and error at most � . As part of our proof we prove upper bounds on the influence of random variables on multivariate functions, which may be of independent interest.
It is known that there exist multi-prover interactive protocols (MIP\mathsf{MIP}MIP protocols) for the complexity class NEXP\mathsf{NEXP}NEXP, succinct MIP\mathsf{MIP}MIP protocols for NP\mathsf{NP}NP and multi-prover interactive protocols with shared entanglement (MIP∗\mathsf{MIP}^\astMIP∗ protocols) for RE\mathsf{RE}RE. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker. In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial ppp, we can construct two-prover one-round MIP\mathsf{MIP}MIP and MIP∗\mathsf{MIP}^\astMIP∗ protocols for NEXP\mathsf{NEXP}NEXP and RE\mathsf{RE}RE, respectively, that are robust against p(n)p(n)p(n) bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round MIP\mathsf{MIP}MIP protocol for NP\mathsf{NP}NP robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.
We introduce the miss-match form for two-prover one-round proof systems. Any two-prover one-round proof system can be easily modified so as to be in miss-match form. Proof systems in miss-match form have the "projection" property that is important for deriving hardness of approximation results for NP-hard combinatorial optimization problems. Our main result is an upper bound on the number of parallel repetitions that suffice in order to reduce the error of miss-match proof systems from p to ϵ\epsilonϵ. This upper bound depends only on p and on ϵ\epsilonϵ (polynomial in 1/(1-p) and in 1/ϵ1/\epsilon1/ϵ). Based on previous work, it follows that for any ϵ>0,\epsilon >0,ϵ>0, NP has two-prover one-round proof systems with logarithmic-sized questions, constant-sized answers, and error at most ϵ\epsilonϵ. As part of our proof we prove upper bounds on the influence of random variables on multivariate functions, which may be of independent interest.