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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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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.
New directproduct testers and 2query PCPs
 IN PROCEEDINGS OF THE FORTYFIRST ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... The “direct product code” of a function f gives its values on all ktuples (f(x1),..., f(xk)). This basic construct underlies “hardness amplification ” in cryptography, circuit complexity and PCPs. Goldreich and Safra [GS00] pioneered its local testing and its PCP application. A recent result by Din ..."
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The “direct product code” of a function f gives its values on all ktuples (f(x1),..., f(xk)). This basic construct underlies “hardness amplification ” in cryptography, circuit complexity and PCPs. Goldreich and Safra [GS00] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [DG08] enabled for the first time testing proximity to this important code in the “listdecoding” regime. In particular, they give a 2query test which works for polynomially small success probability 1/kα, and show that no such test works below success probability 1/k. Our main result is a 3query test which works for exponentially small success probability exp(−kα). Our techniques (based on recent simplified decoding algorithms for the same code [IJKW08]) also allow us to considerably simplify the analysis of the 2query test of [DG08]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of k, and success probability 1/kα. Finally we show the applicability of the new tests to PCPs. Starting with a 2query PCP over an alphabet Σ and with soundness error 1 − δ, Rao [Rao08] (building on Raz’s (kfold)
The Structure of Winning Strategies in Parallel Repetition Games
"... Abstract. Given a function f: X! Σ, its ℓwise direct product is the function F = f ℓ: Xℓ! Σℓ dened by F (x1,..., xℓ) = (f(x1),..., f(xℓ)). A two prover game G is a game that involves 3 participants: V,A, and B. V picks a random pair (x, y) and sends x to A, and y to B. A responds with f(x), B with ..."
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Abstract. Given a function f: X! Σ, its ℓwise direct product is the function F = f ℓ: Xℓ! Σℓ dened by F (x1,..., xℓ) = (f(x1),..., f(xℓ)). A two prover game G is a game that involves 3 participants: V,A, and B. V picks a random pair (x, y) and sends x to A, and y to B. A responds with f(x), B with g(y). A,B win if V (x, y, f(x), g(y)) = 1. The repeated game Gℓ is the game where A,B get ℓ questions in a single round and each of them responds with an ℓ symbol string (this is also called the parallel repetition of the game). A,B win if they win each of the questions. In this work we analyze the structure of the provers that win the repeated game with non negligible probability. We would like to deduce that in such a case A,B must have a global structure, and in particular they are close to some direct product encoding. A similar question was studied by the authors and by Impagliazzo et. al. in the context of testing Direct Product. Their result can be be interpreted as follows: For a specic game G, if A,B win Gℓ with non negligible probability, then A,B must be close to be a direct product encoding. We would like to generalize these results for any 2prover game. In this work we prove two main results: In the rst part of the work we show that for a certain type of games, there exist A,B that win the repeated game with non negligible probability yet are still very far from any Direct Product encoding. In contrast, in the second part of the work we show that for a certain type of games, called \miss match " games, we have the following behavior. Whenever A,B win non negligibly then they are both close to a Direct Product strategy.