Robust pcps of proximity, shorter pcps and applications to coding

Abstract

We continue the study of the trade-off between the length of PCP sand their query complexity, establishing the following main results(which refer to proofs of satisfiability of circuits of size n): 1 We present PCPs of length exp(Õ(log log n)2)•n that can be verified by making o(log logn) Boolean queries.For every ε>0, we present PCPs of length exp(logε n)• n that can be verified by making a constant number of Boolean queries. In both cases, false assertions are rejected withconstant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi-polylogarithmic in the first case (ofquery complexity o(log logn)), and 2(log n)ε, for any ε>0, in the second case (of constant query complexity). In contrast, previous results required at least 2 √logn overhead in the length, even to get query complexity 2 √log n. Our techniques include the introduction of a new variant of PCPs that we call "Robust PCPs". These new PCPs facilitate proof composition, which is a central ingredient in construction of PCP systems. (A related notion and its composition properties were discovered independently by Dinur and Reingold. ) Our main technical contribution is a construction of a "length-efficient" Robust PCP. While the new construction uses many of the standard techniques in PCPs, it does differ from previous constructions in fundamental ways, and in particular does not use the "parallelization" step of Arora et al. . The alternative approach may be of independent interest. We also obtain analogous quantitative results for locally testable codes. In addition, we introduce a relaxed notion of locally decodable codes,and present such codes mapping k information bits to code words of length κ1+ε, for any ε>0.