A Combinatorial Consistency Lemma with Application to Proving the PCP Theorem

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 lowdegree 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).