A Sub-constant Error-probability Pcp Characterization of Np Part Ii: the Consistency Test

This paper introduces a new consistency-test for a class of codes, referred to as geometric-codes, and proves the test to be of small error-probability. This consistency-test enables us to conclude a strong characterization of NP in terms of PCP. Speciically, our theorem states that, for any given > 0, membership in any NP language can be veriied with O(1) accesses, each reading logarithmic number of bits, and such that the error-probability is as low as 2 ? log 1? n. Our results are in fact stronger, as stated in Appendix 2. Previous characterizations of NP in terms of PCP have managed to achieve, with constant number of accesses, error-probability of, at best, a constant. Low-degree polynomials are a special case of geometric-codes, hence our consistency-test implies a low-degree{test, which is the rst to exhibit sub-constant probability of error. The proof for the small error-probability consistency-test is, nevertheless, simpler (combinatorial and geometrical in nature) than previous proofs, which achieved only constant probability of error. In Part I of this work, a complete exposition of the proof for the improved PCP theorem is given 1. The exposition there starts oo with the general scheme of AS92], which is altered, using several new ideas as well as known ones, so as to take advantage of the test described herein to obtain the results claimed.