lect18 - 6.841 Advanced Complexity Theory Lecture 18...

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6.841 Advanced Complexity Theory April 18, 2007 Lecture 18 Lecturer: Madhu Sudan Scribe: Nadia Benbernou 1 Probabilistically Checkable Proofs (PCP) The goal of a probabilistically checkable proof is to verify a proof by looking at only a small number of bits, and probabilistically decide whether to accept or reject. The two resources which PCPs rely on are randomness and queries. A Restricted ( r ( n ) , q ( n ) , a ( n )) PCP verifier is a probabilistic polynomial time verifier with oracle access to a proof π that 1. uses at most r ( n ) random bits. 2. makes q ( n ) queries to π . 3. expects answers of size a ( n ). Definition 1 A language L is in PCP s [ r ( n ) , q ( n ) , a ( n )] if there exists a re- stricted ( r ( n ) , q ( n ) , a ( n )) - PCP verifier V with soundness parameter s such that: x L ⇒ ∃ π s.t. Pr[ V π ( x ) accepts ] = 1 x 6∈ L ⇒ ∀ π, Pr[ V π ( x ) accepts ] < s. Theorem 2 (PCP Theorem) There exists a global constant Q such that L NP there is a constant c such that L PCP 1 / 2 [ c log n, Q, 2] The PCP Theorem was first proved by Arora, Safra, Arora, Lund, Motwani, Sudan, and Szegedy, then later by Dinur. It is easy to see that NP = c N PCP 0 [0 , n c , 2], since the proof of membership to a language in NP is polynomial in size, so can just query entire proof and then accept or reject. We also have NP = c N PCP 1 / 2 [ c log n, n c , 2], to see the forward inclusion that NP ⊆ ∪ c N PCP 1 / 2 [ c log n, n c , 2] can simulate log n bits of randomness (just enumerate over all random strings, count the number of accepting and rejecting configurations, and then output decision). 2 MAX-k-SAT Definition 3 The promise problem MAX-k-SAT is given by: Π Yes = { φ | φ is a k -cnf formula and all clauses can be simulaneously satisfied } Π No = { φ | φ is a k -cnf formula and any assignment satisfies < 1 - fraction of the clauses } 18-1
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Corollary 4 There are constants k, > 0 such that MAX - k - SAT is NP-hard to approximate within 1 - .
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  • Spring '09
  • MadhuSudan
  • Computational complexity theory, unsat, PCP theorem, Probabilistically checkable proof, G. Define UNSAT

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