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Uniform direct product theorems: Simplified, unified and derandomized
, 2007
"... The classical DirectProduct Theorem for circuits says that if a Boolean function f: {0, 1} n → {0, 1} is somewhat hard to compute on average by small circuits, then the corresponding kwise direct product function f k (x1,..., xk) = (f(x1),..., f(xk)) (where each xi ∈ {0, 1} n) is significantly ha ..."
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Cited by 19 (4 self)
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The classical DirectProduct Theorem for circuits says that if a Boolean function f: {0, 1} n → {0, 1} is somewhat hard to compute on average by small circuits, then the corresponding kwise direct product function f k (x1,..., xk) = (f(x1),..., f(xk)) (where each xi ∈ {0, 1} n) is significantly harder to compute on average by slightly smaller circuits. We prove a fully uniform version of the DirectProduct Theorem with informationtheoretically optimal parameters, up to constant factors. Namely, we show that for given k and ɛ, there is an efficient randomized algorithm A with the following property. Given a circuit C that computes f k on at least ɛ fraction of inputs, the algorithm A outputs with probability at least 3/4 a list of O(1/ɛ) circuits such that at least one of the circuits on the list computes f on more than 1 − δ fraction of inputs, for δ = O((log 1/ɛ)/k); moreover, each output circuit is an AC 0 circuit (of size poly(n, k, log 1/δ, 1/ɛ)), with oracle access to the circuit C. Using the GoldreichLevin decoding algorithm [GL89], we also get a fully uniform version of Yao’s XOR Lemma [Yao82] with optimal parameters, up to constant factors. Our results simplify and improve those in [IJK06]. Our main result may be viewed as an efficient approximate, local, listdecoding algorithm for
Parallel Repetition of Entangled Games ∗
"... We consider oneround games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of provers? Classically, th ..."
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Cited by 16 (3 self)
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We consider oneround games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of provers? Classically, this question, open for many years, has culminated in Raz’s celebrated parallel repetition theorem on one hand, and in efficient product testers for PCPs on the other. In the case where provers share entanglement, the only previously known results are for special cases of games, and are based on techniques that seem inherently limited. Here we show for the first time that the maximum success probability of entangled games can be reduced through parallel repetition, provided it was not initially 1. Our proof is inspired by a seminal result of Feige and Kilian in the context of classical twoprover oneround interactive proofs. One of the main components in our proof is an orthogonalization lemma for operators, which might be of independent interest. Twoprover games play a major role both in theoretical computer science, where they led to many breakthroughs such as the discovery of tight inapproximability results, and in quantum physics, where they first arose in the context of Bell inequalities. In such games, a referee chooses a pair of questions
Derandomized Parallel Repetition of Structured PCPs
"... Abstract—A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof, and in return is allowed to err with some bounded probability. The probability that the verifier accepts a false proof is cal ..."
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Abstract—A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof, and in return is allowed to err with some bounded probability. The probability that the verifier accepts a false proof is called the soundness error, and is an important parameter of a PCP system that one seeks to minimize. Constructing PCPs with subconstant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability. In this work we construct such PCP verifiers, i.e., PCPs that make only two queries and have subconstant soundness error. Our construction can be viewed as a combinatorial alternative to the “manifold vs. point ” construction, which is the only construction in the literature for this parameter range. The “manifold vs. point ” PCP is based on a low degree test, while our construction is based on a direct product test. Our construction of a PCP is based on extending the derandomized direct product test of Impagliazzo, Kabanets and Wigderson (STOC 09) to a derandomized parallel repetition theorem. More accurately, our PCP construction is obtained in two steps. We first prove a derandomized parallel repetition theorem for specially structured PCPs. Then, we show that any PCP can be transformed into one that has the required structure, by embedding it on a deBruijn graph.