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

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Presentation on theme: "Robust PCPs of Proximity (Shorter PCPs, applications to Coding)"— Presentation transcript:

1 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)

2 PCP Theorem [AS ’92, ALMSS ’92]
x 2 L ? x 2 L ? V V PCP Theorem (deterministic verifier) (probabilistic verifier) Parameters: # queries - constant proof size - polynomial NP Proof Completeness: x 2 L ) 9 ; P r [ V ( = 1 ] Soundness: x = 2 L ) 8 ; P r [ V ( 1 ]

3 Short PCPs? How long is the new PCP? Old NP proof – n ; New PCP - ?

4 Short PCPs vs Query Complexity
queries Blowup in proof size |PCP proof|/|NP proof| [PS ’94] O ( 1 ) n [GS ’02, BSVW ’03] O ( 1 ) 2 p l o g n This paper O ( 1 ) 2 ( l o g n ) Previous PCPs required blowup factor of even when reading bit-locations 2 p l o g n 2 p l o g n o ( l g n ) q u a s i p o l y g n

5 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

6 Proof Techniques

7 Proof Overview New Definition: Robust PCP of Proximity
New Composition Theorem Simple, modular Avoid overhead present in earlier compositions Building Block

8 Robust PCP of Proximity and Composition Theorem

9 Why Composition? Verifier V V ©V
Don’t know to build PCPs with q = O(1) and 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

10 Proof Composition, a la [AS ‘92]
x DR Local Check VL R a n d o m c i s - a 1 2 : Q R q = p o l y g n Completeness: x 2 L ) 9 ; P r [ V ( = 1 ] x 2 L ) 9 ; P r [ D R ( a 1 : Q = ] Soundness: x = 2 L ) 8 ; P r [ V ( 1 ] x = 2 L ) 8 ; P r [ D R ( a 1 : Q ] Need to verify that satisfy local check DR a 1 2 : Q R Idea : Use a PCP verifier to check !

11 Proof Composition, Contd
DR Local Check Create language Check if using a PCP veriifier L R = f ( a 1 ; : Q ) j D c e p t s g ( a 1 ; : Q R ) 2 L a 1 2 : Q R Problem: PCP verifier VLR needs to read all of theorem (input) Idea: Define a new Verifier that “barely reads” the theorem x VL VLR R

12 PCP of Proximity (PCPP) New! (aka, Assignment testers [DR ’03] )
Completeness: x - T h e o r m x 2 L ) 9 ; P r [ V ( = 1 ] Soundness: x = 2 L ) 8 ; P r [ V ( 1 ] x i s f a r o m L ) 8 ; P [ V ( = 1 ] 2 V Important: # queries = sum of queries into theorem + proof Theorem is not encoded in any error correcting code Specialization of “PCP spot checkers” [EKR ’99] (probabilistic verifier)

13 Composition again VL x ¼ x = 2 L ) 8 ¼ ; P r [ ( a : ] > x 2 L ) 9
1 2 : Q R x VLR VL R Completeness: x 2 L ) 9 ; P r [ ( a 1 : Q R ] = Soundness: x = 2 L ) 8 ; P r [ ( a 1 : Q R ] > Problem: Need to distinguish between & PCPP distinguishes between & ( a 1 ; : Q R ) 2 L ( a 1 ; : Q R ) = 2 L ( a 1 ; : Q R ) 2 L f a r o m L R Strengthen soundness condition of verifier VL

14 Robust PCP of Proximity (Robust-PCPP) New!  x x 2 L ) 9 ¼ ; P r [ D (
Local Check V DR a 1 2 : Q R Completeness: x 2 L ) 9 ; P r [ D R ( a 1 : Q = ] Robust Soundness: Soundness: x i s f a r o m L ) 8 ; P [ D R ( 1 : Q = ] > 2 x i s f a r o m L ) 8 ; P [ ( 1 : Q R ] > 2

15 New PCPP Proof for VCOMP = (, R1,….., Rm)
Composition Theorem VIN x R1 VOUT VIN Rm VOUT VIN = VCOMP Req. of Inner Verifier: proximity of inner< robustness of outer Randomness: rCOMP = rOUT + rIN Queries: qCOMP = qIN Proximity: COMP = OUT Robustness: COMP = IN New PCPP Proof for VCOMP = (, R1,….., Rm)

16 Summarizing... n ¢ p o l y g Defined Robust-PCPs of Proximity
Proved a natural composition theorem for robust-PCPPs Simpler and shorter constructions of PCPs Open Questions: Constant query PCPs of size ? n p o l y g

17 The End


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