Robust PCPs of Proximity (Shorter PCPs, applications to Coding) V V V

Robust PCPs of Proximity (Shorter PCPs, applications to Coding) V V V www.phwiki.com

Robust PCPs of Proximity (Shorter PCPs, applications to Coding) V V V

Carroll, Larry, Contributor has reference to this Academic Journal, PHwiki organized this Journal Robust PCPs of Proximity (Shorter PCPs, applications to Coding) Eli Ben-Sasson (Radcliffe) Oded Goldreich (Weizmann & Radcliffe) Prahladh Harsha (MIT) Madhu Sudan (MIT & Radcliffe) Salil Vadhan (Harvard & Radcliffe) PCP Theorem [AS ’92, ALMSS ’92] V (deterministic verifier) V (probabilistic verifier) PCP Theorem NP Proof Completeness: Soundness: Parameters: queries – constant proof size – polynomial Short PCPs How long is the new PCP Old NP proof – n ; New PCP –

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Short PCPs vs Query Complexity Previous PCPs required blowup factor of even when reading bit-locations Why Short PCPs Negative Consequences Tightness of inapproximability results with respect to running time Positive Consequences Future “practical implementations” of proof-verification Coding Theory Locally testable codes [GS ’02, BSVW ’03, this paper] “Relaxed Locally Decodable Codes” [this paper] Cryptography e.g.: non-blackbox techniques Proof Techniques

Proof Overview New Definition: Robust PCP of Proximity New Composition Theorem Simple, modular Avoid overhead present in earlier compositions Building Block Robust PCP of Proximity in addition to Composition Theorem Why Composition Don’t know to build PCPs with q = O(1) in addition to size = poly(n) directly However, [BFLS ’91] type of PCP: size = poly(n ) q = poly log n Verifier V [AS ’92, ALMSS ’92] “magically compose” verifier V with itself to obtain new verifier V ©V with following parameters size = poly(n ) q = poly log log n V ©V

Proof Composition, a la [AS ‘92] VL Completeness: Soundness: DR Local Check Need to verify that satisfy local check DR Idea : Use a PCP verifier to check ! Proof Composition, Contd DR Local Check Create language Check if using a PCP veriifier VL VLR Problem: PCP verifier VLR needs to read all of theorem (input) Idea: Define a new Verifier that “barely reads” the theorem PCP of Proximity (PCPP) New! (aka, Assignment testers [DR ’03] ) V (probabilistic verifier) Completeness: Soundness: Important: queries = sum of queries into theorem + proof Theorem is not encoded in any error correcting code Specialization of “PCP spot checkers” [EKR ’99]

Composition again VL VLR Completeness: Soundness: Problem: Need to distinguish between & PCPP distinguishes between & Strengthen soundness condition of verifier VL PCP of Proximity V Completeness: Soundness: DR Local Check Robust Soundness: (Robust-PCPP) New! Robust Composition Theorem VOUT VIN R1 Rm New PCPP Proof as long as VCOMP = (, R1, , Rm) VOUT + VIN = VCOMP R in addition to omness: rCOMP = rOUT + rIN Robustness: COMP = IN Proximity: COMP = OUT Queries: qCOMP = qIN VIN Req. of Inner Verifier: proximity of inner< robustness of outer Summarizing Defined Robust-PCPs of Proximity Proved a natural composition theorem as long as robust-PCPPs Simpler in addition to shorter constructions of PCPs Open Questions: Constant query PCPs of size The End Carroll, Larry Filmstew.com Contributor www.phwiki.com

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