M. BELLARE, S. GOLDWASSER, C. LUND, AND A. RUSSELL, E#cient probabilistic checkable proofs and applications to approximation, in Proc. STOC 1993, ACM, New York, pp. 294-- 304.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....it follows from [7, 37, 25] that NEXPTIME = MIP(2, 1) with exponentially small probability of error. MIP(2, 1) proof systems have cryptographic applications (see [12, 13, 36, 18] and have also been used as a starting point to prove that certain optimization problems are hard to approximate (see [22, 4, 5, 25, 6, 10, 38, 8]) For more discussion about the class MIP(2, 1) and its applications, see [23] and the references there. Sequential repetition of MIP(2, 1) proof systems decreases the probability of error exponentially, but requires multiple rounds. Parallel repetition preserves the number of rounds. At what ....

....in analyzing the probability of error of a parallel repetition, not only as a mathematical problem, but also because an e#cient technique to decrease the probability of error was needed. In the literature there are many results that use di#erent techniques to decrease the probability of error [19, 33, 37, 25, 8, 23, 43]. For certain applications, however, these techniques are insu#cient. Parallel repetition was suggested as a technique to decrease the probability of error, because it was believed to be very e#cient, and because it preserves many canonical properties of the proof system (e.g. zero knowledge) ....

M. BELLARE, S. GOLDWASSER, C. LUND, AND A. RUSSELL, E#cient probabilistic checkable proofs and applications to approximation, in Proc. STOC 1993, ACM, New York, pp. 294-- 304.

....it follows from [7, 37, 25] that NEXPTIME = MIP(2, 1) with exponentially small probability of error. MIP(2, 1) proof systems have cryptographic applications (see [12, 13, 36, 18] and have also been used as a starting point to prove that certain optimization problems are hard to approximate (see [22, 4, 5, 25, 6, 10, 38, 8]) For more discussion about the class MIP(2, 1) and its applications, see [23] and the references there. Sequential repetition of MIP(2, 1) proof systems decreases the probability of error exponentially, but requires multiple rounds. Parallel repetition preserves the number of rounds. At what ....

....in analyzing the probability of error of a parallel repetition, not only as a mathematical problem, but also because an e#cient technique to decrease the probability of error was needed. In the literature there are many results that use di#erent techniques to decrease the probability of error [19, 33, 37, 25, 8, 23, 43]. For certain applications, however, these techniques are insu#cient. Parallel repetition was suggested as a technique to decrease the probability of error, because it was believed to be very e#cient, and because it preserves many canonical properties of the proof system (e.g. zero knowledge) ....

M. BELLARE, S. GOLDWASSER, C. LUND, AND A. RUSSELL, E#cient probabilistic checkable proofs and applications to approximation, in Proc. STOC 1993, ACM, New York, pp. 294-- 304.

....Satisfiability, Independent Set [2, 3] Clique [13] and Colorability [17] For example, the MAX SAT function maps a Boolean formula in 3 conjunctive normal form to the maximum number of clauses of that formula that are simultaneously satisfied by some assignment to the variables. Bellare et al. [5] showed that there is no polynomial time algorithm that can approximate MAX SAT within ratio 112 113, unless NP = P. More recently, Bellare and Sudan [6] improved 112 113 to 64 65, but their assumption is weaker than NP = P. Similarly, the result that PCD(log n, 1) PSPACE yields ....

M. Bellare, S. Goldwasser, C. Lund, and A. Russell, E#cient probabilistic checkable proofs and applications to approximation, in Proc. 25th Symposium on Theory of Computing, ACM, New York, 1993, pp. 286--293.

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