Small PCPs with low query complexity

Abstract

Most known constructions of probabilistically checkable proofs (PCPs) either blow up the proofsize by a large polynomial, or have a high (though constant) query complexity. In this thesis we give a transformation with slightly-super-cubic blowup in proof-size and a low query complexity.Specifically, the verifier probes the proof in 16 bits and rejects every proof of a false assertion with probability arbitrarily close to 1, while accepting corrects proofs of theorems with probability one.The proof is obtained by revisiting known constructions and improving numerous components therein. In the process we abstract a number of new modules that may be of use in other PCP constructions.

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

“…The initial PCP. Our new proof of Theorem 1.1 modifies the constructions of Polishchuk and Spielman [44] and Harsha and Sudan [31]. The latter construction was already improved in [28,12] to reduce the length of PCPs to n·2Õ ( √ log n) .…”

“…Theorem 3.1 is proved by modifying a construction that establishes Theorem 1.1. We follow [31] and modify their construction. (An alternative approach would be to start from [44], but that construction does not seem amenable to achieving robust soundness.)…”

“…(An alternative approach would be to start from [44], but that construction does not seem amenable to achieving robust soundness.) The construction of [31] may be abstracted as follows: To verify the satisfiability of a circuit of size n, a verifier expects oracles F i : F m → F , i ∈ {1. . .…”

Sound 3-Query PCPPs Are Long

“…Numerous other works, such as Guruswami et al [1998], Håstad and Khot [2005], Samorodnitsky and Trevisan [2000], Engebretsen and Holmerin [2008], Khot and Saket [2006], and Samorodnitsky and Trevisan [2006], to name a few, investigate optimal or nearly optimal trade-offs between the three parameters of query complexity, completeness and soundness, while settling for polynomial length proofs. A different line of research focused on optimizing the trade-off between proof length and query complexity [Polishchuk and Spielman 1994;Harsha and Sudan 2000;Goldreich and Sudan 2006;, 2006Ben-Sasson and Sudan 2008;Dinur 2007;Raz 2008a, 2007] and all of these constructions obtain perfect completeness. Several of these works, most notably Harsha and Sudan [2000], Goldreich and Sudan [2006], Moshkovitz and Raz [2008a], and Moshkovitz and Raz [2007], also strive to simultaneously optimize the fourth parameter, soundness, but have stopped short of constructing a "super-PCP.…”

“…A different line of research focused on optimizing the trade-off between proof length and query complexity [Polishchuk and Spielman 1994;Harsha and Sudan 2000;Goldreich and Sudan 2006;, 2006Ben-Sasson and Sudan 2008;Dinur 2007;Raz 2008a, 2007] and all of these constructions obtain perfect completeness. Several of these works, most notably Harsha and Sudan [2000], Goldreich and Sudan [2006], Moshkovitz and Raz [2008a], and Moshkovitz and Raz [2007], also strive to simultaneously optimize the fourth parameter, soundness, but have stopped short of constructing a "super-PCP. "…”