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NonDeterministic Exponential Time has TwoProver Interactive Protocols
"... We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language ..."
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Cited by 402 (40 self)
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We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful 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 coNPcomplete languages do not admit such proof systems. In sharp contrast, we show that the class of languages having twoprover 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, provably 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 polynomial size circuits then EXP = 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.
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 307 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
Algebraic Methods for Interactive Proof Systems
, 1990
"... We present a new algebraic technique for the construction of interactive proof systems. We use our technique to prove that every language in the polynomialtime hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP=PSPACE (Shamir) and that MIP ..."
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Cited by 305 (29 self)
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We present a new algebraic technique for the construction of interactive proof systems. We use our technique to prove that every language in the polynomialtime hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP=PSPACE (Shamir) and that MIP=NEXP (Babai, Fortnow and Lund).
Efficient probabilistically checkable proofs and applications to approximation
 In Proceedings of STOC93
, 1993
"... 1 ..."
On the Power of MultiProver Interactive Protocols
 Theoretical Computer Science
, 1988
"... this paper we consider a further generalization of the proof system model, due to BenOr, 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 crim ..."
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Cited by 132 (9 self)
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this paper we consider a further generalization of the proof system model, due to BenOr, 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 NPcomplete then the polynomial time hierarchy collapses. Multipleprover 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 multiprover 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...
BPP has Subexponential Time Simulations unless EXPTIME has Publishable Proofs (Extended Abstract)
, 1993
"... ) L'aszl'o Babai Noam Nisan y Lance Fortnow z Avi Wigderson University of Chicago Hebrew University Abstract We show that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time ffl collapses to the second level of the polynomialtime hierarchy, ..."
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Cited by 112 (9 self)
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) L'aszl'o Babai Noam Nisan y Lance Fortnow z Avi Wigderson University of Chicago Hebrew University Abstract We show that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time ffl collapses to the second level of the polynomialtime hierarchy, ffl has polynomialsize circuits and ffl has publishable proofs (EXPTIME=MA). We also show that BPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we show BPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in MA n P . The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random selfreducibility via low degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not ...
Checking the Correctness of Memories
 Algorithmica
, 1995
"... We extend the notion of program checking to include programs which alter their environment. In particular, we consider programs which store and retrieve data from memory. The model we consider allows the checker a small amount of reliable memory. The checker is presented with a sequence of reques ..."
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Cited by 97 (11 self)
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We extend the notion of program checking to include programs which alter their environment. In particular, we consider programs which store and retrieve data from memory. The model we consider allows the checker a small amount of reliable memory. The checker is presented with a sequence of requests (online) to a data structure which must reside in a large but unreliable memory. We view the data structure as being controlled by an adversary. We want the checker to perform each operation in the input sequence using its reliable memory and the unreliable data structure so that any error in the operation of the structure will be detected by the checker with high probability. We present checkers for various data structures. We prove lower bounds of log n on the amount of reliable memory needed by these checkers where n is the size of the structure. The lower bounds are information theoretic and apply under various assumptions. We also show timespace tradeoffs for checking random access memories as a generalization of those for coherent functions. 1
Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 72 (11 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
Arithmetization: A New Method In Structural Complexity Theory
, 1991
"... . We introduce a technique of arithmetization of the process of computation in order to obtain novel characterizations of certain complexity classes via multivariate polynomials. A variety of concepts and tools of elementary algebra, such as the degree of polynomials and interpolation, becomes there ..."
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Cited by 46 (9 self)
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. We introduce a technique of arithmetization of the process of computation in order to obtain novel characterizations of certain complexity classes via multivariate polynomials. A variety of concepts and tools of elementary algebra, such as the degree of polynomials and interpolation, becomes thereby available for the study of complexity classes. The theory to be described provides a unified framework from which powerful recent results follow naturally. The central result is a characterization of ]P in terms of arithmetic straight line programs. The consequences include a simplified proof of Toda's Theorem that PH ` P ]P ; and an infinite class of natural and potentially inequivalent functions, checkable in the sense of Blum et al. Similar characterizations of PSPACE are also given. The arithmetization technique was independently discovered by Adi Shamir. While this simultaneous discovery was driven by applications to interactive proofs, the present paper demonstrates the applicabil...
Quantum property testing
 In Proceedings of 14th SODA
, 2003
"... Hein R"ohrig \Lambda \Lambda ..."