Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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Probabilistically Checkable Proofs (PCPs) provide a format of rewriting and verifying mathematical proofs that allow efficient probabilistic verification based on probing very few bits of the rewritten proof. The celebrated PCP Theorem asserts that probing a constant number of bits suffices (in fact just 3 bits suffice). A natural question that arises in the construction of PCPs is by how much does this encoding blow up the original proof while retaining low query complexity. 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): (1) We present PCPs of length exp(o(log log n) 2) · n that can be verified by making o(log log n) Boolean queries. (2) 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 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 2logn 3 , for any ϵ 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. Our main technical contribution is a construction of a “length-efficient” Robust PCP of proximity. 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 k1+ϵ, for any ϵ 0. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)