A combinatorial consistency lemma with application to proving the PCP Theorem

Webpage for a paper by Oded Goldreich and Muli Safra


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 remove the need for 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).

Material available on-line

Back to either Oded Goldreich's homepage. or general list of papers.