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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.
Algorithms for Boolean function query properties
 SIAM J. COMPUT
, 2003
"... We present new algorithms to compute fundamental properties of a Boolean function given in truthtable form. Specifically, we give an O(N 2.322 log N) algorithm for block sensitivity, an O(N 1.585 log N) algorithm for ‘tree decomposition, ’ and an O(N) algorithm for ‘quasisymmetry.’ These algorithm ..."
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Cited by 6 (4 self)
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We present new algorithms to compute fundamental properties of a Boolean function given in truthtable form. Specifically, we give an O(N 2.322 log N) algorithm for block sensitivity, an O(N 1.585 log N) algorithm for ‘tree decomposition, ’ and an O(N) algorithm for ‘quasisymmetry.’ These algorithms are based on new insights into the structure of Boolean functions that may be of independent interest. We also give a subexponentialtime algorithm for the spacebounded quantum query complexity of a Boolean function. To prove this algorithm correct, we develop a theory of limitedprecision representation of unitary operators, building on work of Bernstein and Vazirani.