## On the Power of Multi-Prover Interactive Protocols (1988)

Venue: | Theoretical Computer Science |

Citations: | 131 - 9 self |

### BibTeX

@INPROCEEDINGS{Fortnow88onthe,

author = {Lance Fortnow and John Rompel and Michael Sipser},

title = {On the Power of Multi-Prover Interactive Protocols},

booktitle = {Theoretical Computer Science},

year = {1988},

pages = {156--161}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper we consider a further generalization of the proof system model, due to Ben-Or, Goldwasser, Kilian and Wigderson [6], where instead of a single prover there may be many. This apparently gives the model additional power. The intuition for this may be seen by considering the case of two criminal suspects who are under interrogation to see if they are guilty of together robbing a bank. Of course they (the provers) are trying to convince Scotland Yard (the verifier) of their innocence. Assuming that they are in fact innocent, it is clear that their ability to convince the police of this is enhanced if they are questioned in separate rooms and can corroborate each other's stories without communicating. We shall see later in this paper that this sort of corroboration is the key to the additional power of multiple provers. Interactive proof systems have seen a number of important applications to cryptography [23, 22], algebraic complexity [3], program testing [7, 8] and distributed computation [16, 23]. For example, a chain of results concerning interactive proof systems [22, 3, 24, 9] conclude that if the graph isomorphism problem is NP-complete then the polynomial time hierarchy collapses. Multiple-prover interactive proof systems have also seen several important applications including the analysis of program testing [7, 4] and the complexity of approximation algorithms [14, 2, 1]. Brief summary of results: First we give a simple characterization of the power of the multi-prover model in terms of probabilistic oracle Turing machines. Then we show that every language accepted by multiple prover interactive proof systems can be computed in nondeterministic exponential time. Babai, Fortnow and Lund [4] have since shown this bound is tight. We then show results like th...

### Citations

1053 | The Knowledge Complexity of Interactive Proof Systems
- Goldwasser, Micali, et al.
- 1985
(Show Context)
Citation Context ...John Rompel y Michael Sipser z -------------------- Laboratory for Computer Science Massachusetts Institute of Technology Cambridge, MA 02139 1 Introduction Interactive proof systems, as described in =-=[23]-=- and [3], are a model in which a probabilistic polynomial time verifier may interactively ask questions of a prover with unbounded computational power in order to decide the truth of a proposition. Th... |

720 | M.: Proof verification and the hardness of approximation problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...rchy collapses. Multiple-prover interactive proof systems have also seen several important applications including the analysis of program testing [7, 4] and the complexity of approximation algorithms =-=[14, 2, 1]-=-. Brief summary of results: First we give a simple characterization of the power of the multi-prover model in terms of probabilistic oracle Turing machines. Then we show that every language accepted b... |

575 |
Optimization, approximation and complexity classes
- PAPADIMITRIOU, YANNAKAKIS
- 1988
(Show Context)
Citation Context ...prove some grave consequences of clique approximation. Arora and Safra [2] improved these results and Arora, Lund, Motwani, Sudan and Szegedy [1] applied these techniques to the MAX SNP-hard problems =-=[28]-=-. 3 Probabilistic Oracle Machines Suppose a prover in an interactive proof system must set all his possible responses before the protocol with the verifier takes place. We can think of the prover as a... |

406 | Non-deterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1990
(Show Context)
Citation Context ... problem is NP-complete then the polynomial time hierarchy collapses. Multiple-prover interactive proof systems have also seen several important applications including the analysis of program testing =-=[7, 4]-=- and the complexity of approximation algorithms [14, 2, 1]. Brief summary of results: First we give a simple characterization of the power of the multi-prover model in terms of probabilistic oracle Tu... |

379 | Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems
- Goldreich, Micali, et al.
- 1991
(Show Context)
Citation Context ... see later in this paper that this sort of corroboration is the key to the additional power of multiple provers. Interactive proof systems have seen a number of important applications to cryptography =-=[23, 22]-=-, algebraic complexity [3], program testing [7, 8] and distributed computation [16, 23]. For example, a chain of results concerning interactive proof systems [22, 3, 24, 9] conclude that if the graph ... |

367 | Probabilistic checking of proofs; a new characterization of np
- Arora, Safra
- 1992
(Show Context)
Citation Context ...rchy collapses. Multiple-prover interactive proof systems have also seen several important applications including the analysis of program testing [7, 4] and the complexity of approximation algorithms =-=[14, 2, 1]-=-. Brief summary of results: First we give a simple characterization of the power of the multi-prover model in terms of probabilistic oracle Turing machines. Then we show that every language accepted b... |

309 |
Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes
- Babai, Moran
- 1988
(Show Context)
Citation Context ...t if the provers can be prevented from communicating among themselves through the protocol then parallel runs of the protocol work independently like parallel runs of one prover interactive protocols =-=[5]-=-. Unfortunately, this assumption is fallacious. As mentioned in section 2, results of Cai, Condon, Lipton, Lapidot, Shamir, Feige and Lov'asz [12, 10, 11, 13, 26, 15] show that we can create a two-pro... |

308 | Algebraic methods for interactive proof systems
- Lund, Fortnow, et al.
- 1992
(Show Context)
Citation Context ...25]. Subsequent to the results described in this paper, the complexity of interactive proof systems have been shown to be much more powerful then previously believed. Lund, Fortnow, Karloff and Nisan =-=[27]-=- have shown the existence of an interactive proof system for every language in the polynomial time hierarchy. Using the techniques of Lund, Fortnow, Karloff and Nisan, Shamir [29] has shown that every... |

306 | Programs That Check Their Work
- Blum, Kannan
- 1989
(Show Context)
Citation Context ...tion is the key to the additional power of multiple provers. Interactive proof systems have seen a number of important applications to cryptography [23, 22], algebraic complexity [3], program testing =-=[7, 8]-=- and distributed computation [16, 23]. For example, a chain of results concerning interactive proof systems [22, 3, 24, 9] conclude that if the graph isomorphism problem is NP-complete then the polyno... |

302 | Trading group theory for randomness
- Babai
- 1985
(Show Context)
Citation Context ...el y Michael Sipser z -------------------- Laboratory for Computer Science Massachusetts Institute of Technology Cambridge, MA 02139 1 Introduction Interactive proof systems, as described in [23] and =-=[3]-=-, are a model in which a probabilistic polynomial time verifier may interactively ask questions of a prover with unbounded computational power in order to decide the truth of a proposition. This is a ... |

185 |
Approximating clique is almost NPcomplete
- Feige, Goldwasser, et al.
- 1991
(Show Context)
Citation Context ...rchy collapses. Multiple-prover interactive proof systems have also seen several important applications including the analysis of program testing [7, 4] and the complexity of approximation algorithms =-=[14, 2, 1]-=-. Brief summary of results: First we give a simple characterization of the power of the multi-prover model in terms of probabilistic oracle Turing machines. Then we show that every language accepted b... |

184 |
Private coins versus public coins in interactive proof systems, in: Randomness and Computation
- Goldwasser, Sipser
- 1989
(Show Context)
Citation Context ...tant applications to cryptography [23, 22], algebraic complexity [3], program testing [7, 8] and distributed computation [16, 23]. For example, a chain of results concerning interactive proof systems =-=[22, 3, 24, 9]-=- conclude that if the graph isomorphism problem is NP-complete then the polynomial time hierarchy collapses. Multiple-prover interactive proof systems have also seen several important applications inc... |

132 | Multi-prover interactive proofs: how to remove intractability assumptions
- Ben-Or, Goldwasser, et al.
(Show Context)
Citation Context ...system in which the verifier may only listen and not speak or toss coins. In this paper we consider a further generalization of the proof system model, due to Ben-Or, Goldwasser, Kilian and Wigderson =-=[6]-=-, where instead of a single prover there may be many. This apparently gives the model additional power. The intuition for this may be seen by considering the case of two criminal suspects who are unde... |

119 |
Does co-NP have short interactive proofs
- Boppana, H˚astad, et al.
- 1987
(Show Context)
Citation Context ...tant applications to cryptography [23, 22], algebraic complexity [3], program testing [7, 8] and distributed computation [16, 23]. For example, a chain of results concerning interactive proof systems =-=[22, 3, 24, 9]-=- conclude that if the graph isomorphism problem is NP-complete then the polynomial time hierarchy collapses. Multiple-prover interactive proof systems have also seen several important applications inc... |

95 | Two-prover one-round proof systems: their power and their problems
- Feige, Lovász
- 1992
(Show Context)
Citation Context ...shown that every language accepted in nondeterministic exponential time has a two-prover interactive proof system. 2 A series of results due to Cai, Condon, Lipton, Lapidot, Shamir, Feige and Lov'asz =-=[12, 10, 11, 13, 26, 15]-=- have modified the protocol of Babai, Fortnow and Lund [4] to show that every language in NEXP has a two-prover, one-round proof systems with an exponentially small error, strengthening the (unproven)... |

87 | The Complexity of Perfect Zero-Knowledge
- Fortnow
- 1987
(Show Context)
Citation Context ...active proof system has a perfect zero-knowledge two prover proof system, where even NP does not have perfect zero-knowledge single prover proof systems unless the polynomial-time hierarchy collapses =-=[17]-=-. They also show two prover systems can simulate any multi-prover system. Along the lines of Furer, Goldreich, Mansour, Sipser and Zachos [21], they show any two prover system has an equivalent system... |

43 |
Fully parallelized multi prover protocols for NEXP-time (extended abstract
- Lapidot, Shamir
- 1991
(Show Context)
Citation Context ...shown that every language accepted in nondeterministic exponential time has a two-prover interactive proof system. 2 A series of results due to Cai, Condon, Lipton, Lapidot, Shamir, Feige and Lov'asz =-=[12, 10, 11, 13, 26, 15]-=- have modified the protocol of Babai, Fortnow and Lund [4] to show that every language in NEXP has a two-prover, one-round proof systems with an exponentially small error, strengthening the (unproven)... |

39 |
Uses of Randomness in Algorithms and Protocols
- Kilian
- 1990
(Show Context)
Citation Context ..., Mansour, Sipser and Zachos [21], they show any two prover system has an equivalent system that accepts with probability one for strings in the language. Complete proofs of these results appeared in =-=[25]-=-. Subsequent to the results described in this paper, the complexity of interactive proof systems have been shown to be much more powerful then previously believed. Lund, Fortnow, Karloff and Nisan [27... |

37 | Are there interactive protocols for co-NP languages
- Fortnow, Sipser
- 1988
(Show Context)
Citation Context ... A. We show that any proof of this result can not relativize: Theorem 5.1 There exists an oracle A and a language L 2 co-NP A such that L 62 MIP A . Theorem 5.1 extends a result by Fortnow and Sipser =-=[20]-=- that shows the existence of an oracle relative to which co-NP does not have single prover interactive proofs. Proof In this proof we will use the oracle machine model. It is easy to verify that the p... |

26 | On Completeness and Soundness in Interactive Proof Systems
- Fürer, Goldreich, et al.
- 1989
(Show Context)
Citation Context ...of systems unless the polynomial-time hierarchy collapses [17]. They also show two prover systems can simulate any multi-prover system. Along the lines of Furer, Goldreich, Mansour, Sipser and Zachos =-=[21]-=-, they show any two prover system has an equivalent system that accepts with probability one for strings in the language. Complete proofs of these results appeared in [25]. Subsequent to the results d... |

20 |
R.: Selftesting and Self-correcting programs, with Applications to Numerical Programs
- Blum, Luby, et al.
- 1990
(Show Context)
Citation Context ...tion is the key to the additional power of multiple provers. Interactive proof systems have seen a number of important applications to cryptography [23, 22], algebraic complexity [3], program testing =-=[7, 8]-=- and distributed computation [16, 23]. For example, a chain of results concerning interactive proof systems [22, 3, 24, 9] conclude that if the graph isomorphism problem is NP-complete then the polyno... |

19 | Complexity-theoretic aspects of interactive proof systems
- Fortnow
- 1989
(Show Context)
Citation Context ...tead of showing how to parallelize general multiprover protocols. We show the parallelization assumption faulty in even a simple case with the following counterexample. This example first appeared in =-=[18]-=-. Suppose we have the following two prover protocol: V : Pick two bits a and b uniformly and independently at random. V!P 1 : a V!P 2 : b P 1 !V : c P 2 !V : d V : Accept if (asc) 6= (bsd). It is easy... |

18 | PSPACE is provable by two provers in one round
- Cai, Condon, et al.
- 1991
(Show Context)
Citation Context ...shown that every language accepted in nondeterministic exponential time has a two-prover interactive proof system. 2 A series of results due to Cai, Condon, Lipton, Lapidot, Shamir, Feige and Lov'asz =-=[12, 10, 11, 13, 26, 15]-=- have modified the protocol of Babai, Fortnow and Lund [4] to show that every language in NEXP has a two-prover, one-round proof systems with an exponentially small error, strengthening the (unproven)... |

14 |
On bounded round multi-prover interactive proof systems
- Cai, Condon, et al.
- 1990
(Show Context)
Citation Context |

8 |
On games of incomplete information
- CAI, CONDON, et al.
- 1992
(Show Context)
Citation Context |

7 |
On the success probability of the two provers in one round proof systems
- Feige
- 1991
(Show Context)
Citation Context |

2 |
From scratch to byzantine agreement in constant expected time
- Feldman, Micali
- 1988
(Show Context)
Citation Context ...ower of multiple provers. Interactive proof systems have seen a number of important applications to cryptography [23, 22], algebraic complexity [3], program testing [7, 8] and distributed computation =-=[16, 23]-=-. For example, a chain of results concerning interactive proof systems [22, 3, 24, 9] conclude that if the graph isomorphism problem is NP-complete then the polynomial time hierarchy collapses. Multip... |