A combinatorial consistency lemma with application to proving the PCP theorem

  • Oded Goldreich
  • Shmuel Safra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)


The current proof of the PCP Theorem (i.e., NP=PCP (log, O(1))) is very complicated. One source of difficulty is the technically involved analysis of low-degree tests. Here, we refer to the difficulty of obtaining strong results regarding low-degree tests; namely, results of the type obtained and used by Arora and Safra and Arora et. al.

In this paper, we eliminate the need to obtain such strong results on low-degree tests when proving the PCP Theorem. Although we do not get rid of low-degree tests altogether, using our results it is now possible to prove the PCP Theorem using a simpler analysis of low-degree tests (which yields weaker bounds). In other words, we replace the strong algebraic analysis of low-degree tests presented by Arora and Safra and Arora et. al. by a combinatorial lemma (which does not refer to low-degree tests or polynomials).


Parallelization of Probabilistic Proof Systems Probabilistically Checkable Proofs (PCP) NP Low-Degree Tests 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Oded Goldreich
    • 1
  • Shmuel Safra
    • 2
  1. 1.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.Computer Science Department, Sackler Faculty of Exact SciencesTel-Aviv UniversityRamat-AvivIsrael

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