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, ContdDR
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 ) 91 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