Eli Ben-Sasson, Oded
Goldreich, Prahladh Harsha, Madhu
We continue the study of the trade-off between the length of
PCPs and their query complexity, establishing the following main results
(which refer to proofs of satisfiability of circuits of size n):
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(loglog n), and 2^{(log n)^\eps, for any \eps >
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 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 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 codewords of length k^{1+\eps}, for
any \eps>0.