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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.
On Pseudorandomness and ResourceBounded Measure
 Theoretical Computer Science
, 1997
"... In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all ff ? 0 we show that if there is a language in E = DTIME(2 O(n) ) that is hard to approximate by nondeterministic circuits of size 2 ffn , then there is a pseudorandom generator that can be u ..."
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Cited by 42 (3 self)
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In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all ff ? 0 we show that if there is a language in E = DTIME(2 O(n) ) that is hard to approximate by nondeterministic circuits of size 2 ffn , then there is a pseudorandom generator that can be used to derandomize BP \Delta NP (in symbols, BP \Delta NP = NP). By applying this extension we are able to answer some open questions in [14] regarding the derandomization of the classes BP \Delta \Sigma P k and BP \Delta \Theta P k under plausible measure theoretic assumptions. As a consequence, if \Theta P 2 does not have pmeasure 0, then AM " coAM is low for \Theta P 2 . Thus, in this case, the graph isomorphism problem is low for \Theta P 2 . By using the NisanWigderson design of a pseudorandom generator we unconditionally show the inclusion MA ` ZPP NP and that MA " coMA is low for ZPP NP . 1 Introduction In recent years, following the development of resourcebounded meas...
Relativizable And Nonrelativizable Theorems In The Polynomial Theory Of Algorithms
 In Russian
, 1993
"... . Starting with the paper of Baker, Gill and Solovay [BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision tree ..."
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Cited by 36 (0 self)
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. Starting with the paper of Baker, Gill and Solovay [BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision trees of a certain type or for machines with polylogarithmic restrictions on time. The following question arises: Are these methods of proving "relativized" results universal? In the first part of the present paper we propose a general framework in which assertions of universality of this kind may be formulated and proved as convenient criteria. Using these criteria we obtain, as easy consequences of the known results on boolean decision trees, some new "relativized" results and new proofs of some known results. In the second part of the present paper we apply these general criteria to many particular cases. For example, for many of the complexity classes studied in the literature all relativiza...
ComplexityTheoretic Aspects of Interactive Proof Systems
, 1989
"... In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have a ..."
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Cited by 19 (3 self)
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In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have an important part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation. This thesis will study the complexity of various models of interactive proof systems. A perfect zeroknowledge interactive protocol convinces a verifier that a string is in a language without revealing any additional knowledge in an information theoretic sense. This thesis will show that for any language that has a perfect zeroknowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zeroknowledge protocols for NPcomplete languages unless the polynomialtime hierarchy collapses. Thus knowledge comp...
Uniform Hardness Versus Randomness Tradeoffs For ArthurMerlin Games
, 2003
"... Impagliazzo and Wigderson proved a uniform hardness vs. ..."
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Cited by 8 (6 self)
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Impagliazzo and Wigderson proved a uniform hardness vs.
Graph Isomorphism is Low for ZPP(NP) and other Lowness results
, 2000
"... We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several grouptheoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets th ..."
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Cited by 7 (0 self)
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We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several grouptheoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets that have interactive proof systems with honest provers in P=poly, is also low for ZPP NP . We consider lowness properties of nonuniform function classes, namely, NPMV=poly, NPSV=poly, NPMV t =poly, and NPSV t =poly. Specifically, we show that { Sets whose characteristic functions are in NPSV=poly and that have program checkers (in the sense of Blum and Kannan [8]) are low for AM and ZPP NP . { Sets whose characteristic functions are in NPMV t =poly are low for p 2 .
New Lowness Results for ZPP^NP and other Complexity Classes
, 2000
"... We show that the class AM\coAM is low for ZPP . As a consequence, it follows that Graph Isomorphism and several grouptheoretic problems are low for ZPP . We also ..."
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Cited by 6 (1 self)
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We show that the class AM\coAM is low for ZPP . As a consequence, it follows that Graph Isomorphism and several grouptheoretic problems are low for ZPP . We also
Uniform hardness vs. randomness tradeoffs for ArthurMerlin games
"... Impagliazzo and Wigderson proved a uniform hardness vs. randomness "gap result" for BPP. We show an analogous result for AM: Either ArthurMerlin protocols are very strong and everything in E = ) can be proved to a subexponential time verifier, or else ArthurMerlin protocols are weak and every la ..."
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Cited by 6 (0 self)
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Impagliazzo and Wigderson proved a uniform hardness vs. randomness "gap result" for BPP. We show an analogous result for AM: Either ArthurMerlin protocols are very strong and everything in E = ) can be proved to a subexponential time verifier, or else ArthurMerlin protocols are weak and every language in AM has a polynomial time nondeterministic algorithm in the uniform averagecase setting (i.e., it is infeasible to come up with inputs on which the algorithm fails). For the class AM coAM we can remove the averagecase clause and show under the same assumption that AM coNP.
Lower Bounds for Swapping Arthur and Merlin ∗
, 2007
"... We prove a lower bound for swapping the order of Arthur and Merlin in tworound MerlinArthur games using blackbox techniques. Namely, we show that any AMgame requires time Ω(t 2) to blackbox simulate MAgames running in time t. Thus, the known simulations of MA by AM with quadratic overhead, dat ..."
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We prove a lower bound for swapping the order of Arthur and Merlin in tworound MerlinArthur games using blackbox techniques. Namely, we show that any AMgame requires time Ω(t 2) to blackbox simulate MAgames running in time t. Thus, the known simulations of MA by AM with quadratic overhead, dating back to Babai’s original paper on ArthurMerlin games, are tight within this setting. The blackbox lower bound also yields an oracle relative to which MATIME[n] � AMTIME[o(n 2)]. Complementing our lower bounds for swapping Merlin in MAgames, we prove a timespace lower bound for simulations that drop Merlin entirely. We show that for any c < √ 2, there exists a positive d such that there is a language recognized by lineartime MAgames with onesided error but not by probabilistic randomaccess machines with twosided error that run in time n c and space n d. This improves recent results that give such lower bounds for problems in the second level of the polynomialtime hierarchy.