Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding

Abstract

We continue the study of the trade‐off between the length of probabilistically checkable proofs (PCPs) and 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 $\exp(o(\log\log n)^2)\cdot n$ that can be verified by making $o(\log\log n)$ Boolean queries. 2. For every \epsilon>0, we present PCPs of length $\exp(\log^\epsilon n)\cdot n$ that can be verified by making a constant number of Boolean queries. In both cases, false assertions are rejected with constant 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 (of query complexity $o(\log\log n)$), and is $2^{(\log n)^\epsilon}$, for any $\epsilon > 0$, in the second case (of constant query complexity). Our techniques include the introduction of a new variant of PCPs that we call “robust PCPs of proximity.” These new PCPs facilitate proof composition, which is a central ingredient in the 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 of proximity. While the new construction uses many of the standard techniques used in PCP constructions, it does differ from previous constructions in fundamental ways, and in particular does not use the “parallelization” step of Arora et al. [J. ACM, 45 (1998), pp. 501–555]. 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 codewords of length $k^{1+\epsilon}$ for any $\epsilon>0$.