TR04-021 Authors: Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan

Publication: 24th March 2004 08:29

Downloads: 2209

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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 $n$):

We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$

that can be verified by making $o(\log\log n)$ Boolean queries.

For every $\e>0$, we present PCPs of length $\exp(\log^\e n)\cdot n$

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 $2^{(\log n)^\epsilon}$,

for any $\epsilon > 0$, in the second case (of constant query complexity).

In contrast, previous results required at least $2^{\sqrt{\log n}}$

overhead in the length, even to get query complexity $2^{\sqrt{\log n}}$.

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 $k^{1+\e}$, for any $\e>0$.