11

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

Eli Ben-Sasson

(Radcliffe)

Oded Goldreich(Weizmann & Radcliffe)

Prahladh Harsha(MIT)

Madhu Sudan(MIT & Radcliffe)

Salil Vadhan(Harvard & Radcliffe)

22

NP – Classical ProofsNP – Classical Proofs NP – Class of languages that have short proofs of NP – Class of languages that have short proofs of

membershipmembership

V(determinist

icverifier)

Proof

x 2 L

Graph Graph ColoringColoring

Formula ÁGraph G

Satisfiability Proof =3 coloring

Proof =Satisfying

assignment

Completeness:

Soundness:

0

0

1

0

1

0

1

1

x 2 L ) 9¼;V(x;¼) = accept

x =2 L ) 8¼;V(x;¼) = reject

(x _ y_ ¹z) :::(¹x _ y_ z)

¼

33

PCP Theorem [AS ’92, ALMSS ’92]PCP Theorem [AS ’92, ALMSS ’92]

V(determinist

icverifier)

V

(probabilisticverifier)

PCP Theorem

NP Proof

Completeness:

Soundness: x =2 L ) 8¼;Pr[V¼(x) = 1] · 12

x 2 L ) 9¼;Pr[V¼(x) = 1] = 1

¼

Parameters:1. # random coins - O(log n)2. # queries - constant3. proof size - polynomial

44

PCPs - SignificancePCPs - Significance

Major impact on the study of combinatorial Major impact on the study of combinatorial optimization optimization Consequence: For many NP-hard combinatorial Consequence: For many NP-hard combinatorial

optimization problems, finding near-optimal optimization problems, finding near-optimal solutions is also NP-hardsolutions is also NP-hard

Approximating Approximating MAXSATMAXSAT to within a factor of to within a factor of (8/7 - (8/7 - )), for any, for any > 0 > 0,, is NP-hard is NP-hard

(Will not dwell into consequence on combinatorial (Will not dwell into consequence on combinatorial optimization)optimization)

55

Short PCPs?Short PCPs? How long is the new PCP proof?How long is the new PCP proof?

Old NP proof – Old NP proof – nn ; New PCP proof - ; New PCP proof - ??

Why Short PCPs?Why Short PCPs? Upper boundsUpper bounds

Cryptography Cryptography Computationally Sound Proofs and applications Computationally Sound Proofs and applications [Kil [Kil

’92, Mic ’94, CGH ’98, Bar ’01] ’92, Mic ’94, CGH ’98, Bar ’01] Coding TheoryCoding Theory

Locally testable codes Locally testable codes [GS ’02, BSVW ’03, this [GS ’02, BSVW ’03, this paper]paper]

““Relaxed Locally Decodable Codes” Relaxed Locally Decodable Codes” [this paper][this paper]

66

Why Short PCPs? (Contd)Why Short PCPs? (Contd)

Lower BoundsLower Bounds Tightness of approximation algorithms with Tightness of approximation algorithms with

respect to running timerespect to running time e.g.: If SAT has a PCP of size e.g.: If SAT has a PCP of size nn then then

+

SAT 2 TIME¡2 (n)

¢

Approximating requires time at least MAXSAT 2n1=®

77

Short PCPs – Earlier ResultsShort PCPs – Earlier Results

[PS ’94][PS ’94] Proof Size = Proof Size = nn1+1+, query = O(1/, query = O(1/))

(Constant hidden in big-O (Constant hidden in big-O ¼¼ 10 1066 ) )

[Hås ’97][Hås ’97] Proof Size = nProof Size = n10000001000000, query = 3; , query = 3;

88

Short PCPs vs Query ComplexityShort PCPs vs Query Complexityqueriesqueriesproof sizeproof size

[HS ’00][HS ’00]

[GS ’02, BSVW ’03][GS ’02, BSVW ’03]

This paper This paper

n3+²

n ¢2p

logn

n ¢2(logn)²O(1

²)

n ¢2(log logn)c

O(1)

o(loglogn)

17

(= n1+o(1))

99

Our Main ResultsOur Main ResultsMain Theorem:Main Theorem:

Satisfiability of circuits of size Satisfiability of circuits of size nn can be can be probabilistically verifiedprobabilistically verified By probing a proof of length By probing a proof of length

in bit-locations.in bit-locations.

OROR By probing a proof of length By probing a proof of length

in bit-locations.in bit-locations.

Previous PCPs required length proof Previous PCPs required length proof size even when reading bit-locations size even when reading bit-locations

[GS ’02, BSVW ’03][GS ’02, BSVW ’03]

n ¢2(logn)²

O(1²)

n ¢2p

logn

n ¢2(log logn)c

o(loglogn)

2p

logn

1010

Proof TechniquesProof Techniques

New Definition: Robust PCP of New Definition: Robust PCP of Proximity Proximity

New Composition Theorem New Composition Theorem Essential for short PCPsEssential for short PCPs simple, modularsimple, modular

Building BlockBuilding Block

1111

Robust PCP of ProximityRobust PCP of Proximityandand

Composition TheoremComposition Theorem

1212

PCP – Definition (Recall)PCP – Definition (Recall)

VL

(probabilisticverifier)

x - Theorem Completeness:

Soundness:

x 2 L ) 9¼;Pr[V¼(x) = 1] = 1

¼x =2 L ) 8¼;Pr[V¼(x) = 1] · 1

2

Parameters of Interest:• size of proof (|• # queries (q )

|| · q ¢ 2rand

1313

Why Composition?Why Composition?

Don’t know to build PCPs with Don’t know to build PCPs with q = Oq = O(1)(1) and and

sizesize = = poly(poly(nn)) directly directly However,However,

[AS ’92, ALMSS ’92] 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

1414

Proof Composition, a la [AS ‘92]Proof Composition, a la [AS ‘92]

VL

r = O(logn)q= poly logn

¼

x

Completeness:

Soundness:

x 2 L ) 9¼;Pr[V¼(x) = 1] = 1x =2 L ) 8¼;Pr[V¼(x) = 1] · 1

2

DR

ConsistencyCheck

a1 a2 : :: aQR

Need to check if satisfy consistency check DRa1 a2 : : : aQR

Idea : Use a PCP verifier to check !

x 2 L ) 9¼;Pr[DR (a1 : : :aQR) = 1] = 1

x =2 L ) 8¼;Pr[DR (a1 : : :aQR) = 1] · 1

2

Random coins - R

1515

Proof Composition, ContdProof Composition, Contd

DR

ConsistencyCheck

a1 a2 : :: aQR

Create language

Check if using a PCP veriifier

LR = f(a1; : : : ;aQR)jDR accepts (a1; : : : ;aQR

)g

(a1; : : : ;aQR) 2 LR

VL ¼

x

VLR

¼R

Problem: PCP verifier VLR

needs to read all of

theorem (input)Key Observation:• “PCP Verifier barely looks at Theorem”• [BFLS ’91] : Assume theorem is encoded andcount #queries into theorem

1616

[BFLS ’91] PCP Verifier (Holographic [BFLS ’91] PCP Verifier (Holographic Proofs)Proofs)

V

(probabilisticverifier)

x - Theorem Completeness:

Soundness:

¼

Important: • # queries = sum of queries into encoded theorem + proof

E nc(x) - Encoding x 2 L )9¼;Pr[VE nc(x);¼= 1] = 1

y¡ far from Enc(L) )8¼;Pr[Vy;¼ = 1] · 1

2

y

1717

Proof Composition, ContdProof Composition, Contd

VL

x

VLR

¼R

a1 a2 : :: aQR

¼Enc(a1 a2 : : : aQR

)

Problem: Need to check and areconsistent.

Semantics of arranging this is complex.

Earlier performed by“parallelization” – costlyin randomness (large proof size)

(a1; : : : ;aQR)

Enc(a1 a2 : : : aQR)

Idea: Remove restriction that theorem is encoded !

1818

PCP of Proximity (PCPP)PCP of Proximity (PCPP)

V

(probabilisticverifier)

x - Theorem Completeness:

Soundness:

¼

• # queries = sum of queries into theorem + proof • Theorem in un-encoded format• – proximity parameter

x 2 L ) 9¼;Pr[V x;¼= 1] = 1

¢ (x;L) > ±)8¼;Pr[Vx;¼() = 1] · 1

2

x =2 L )8¼;Pr[Vx;¼() = 1] · 1

2

1919

Composition againComposition again

VLVLR

¼R

a1 a2 : :: aQR

¼

x

Completeness:

Soundness:Problem:

Need to distinguish between & PCPP distinguishes between &

(a1; :: : ;aQR) 2 LR (a1; :: : ;aQR

) =2 LR(a1; :: : ;aQR

) 2 LR ±¡ far from LR

Strengthen soundness condition of verifier VL

x 2 L ) 9¼;Pr[(a1; : : : ;aQR) 2 LR ] = 1

x =2 L ) 8¼;Pr[(a1;: : :;aQR) =2 LR ]> 1

2

2020

PCP of ProximityPCP of Proximity

V

Completeness:

Soundness:

DR

ConsistencyCheck

Robust Soundness:

- robustness parameter of robust-PCPP

(Robust-PCPP) (Robust-PCPP) New!New!RobustRobust

x

a1 a2 : aQR

x 2 L ) 9¼;Pr[DR (a1; : : : ;aQR) = 1] = 1

¢ (x;L) > ±) 8¼;Pr[DR (a1; : : : ;aQR) = 1] · 1

2¢ (x;L) > ±)

8¼;Pr[(a1; : : : ;aQR) is ½-far from LR ] > 1

2

2121

Composition TheoremComposition Theorem

VOUT

VIN R1

Rm

New PCPP Proof for VCOMP = (, R1,….., Rm)

VOUT + VIN = VCOMP

Randomness: rCOMP = rOUT + rIN

Robustness: COMP = INProximity: COMP = OUT

Queries: qCOMP = qIN

x

VIN

• Req. of Inner Verifier: IN (proximity) < OUT (robustness)

2222

Advantages of PCPPsAdvantages of PCPPs

Give shortest known PCPsGive shortest known PCPs Allow natural self-compositionAllow natural self-composition Simpler constructions of PCPs (no parallelization)Simpler constructions of PCPs (no parallelization) Coding applications:Coding applications:

Simple, highly efficient Locally Testable CodesSimple, highly efficient Locally Testable Codes Simple, highly efficient Relaxed Locally Simple, highly efficient Relaxed Locally

Decodable CodesDecodable Codes Any efficient property is locally testable (with a Any efficient property is locally testable (with a

little bit of help)little bit of help)

2323

PCPPs – Brief HistoryPCPPs – Brief History

Holographic proofs - PCPPs where assignment Holographic proofs - PCPPs where assignment xx

is encoded. is encoded. [BFLS ’91] [BFLS ’91] PCPP - implicit in low-degree tests PCPP - implicit in low-degree tests

[RS ’92, ALMSS ’92][RS ’92, ALMSS ’92] PCPPs - special case of “PCP Spot Checkers” PCPPs - special case of “PCP Spot Checkers”

[EKR ’99][EKR ’99] PCPP – extension of Property Testing PCPP – extension of Property Testing

[RS ’92, GGR ’96][RS ’92, GGR ’96] Assignment Testers ofAssignment Testers of [DR ’03] [DR ’03] similar to PCPPs.similar to PCPPs.

2424

Building BlockBuilding Block

2525

Robust PCPPs constructionsRobust PCPPs constructions

Most existing PCP constructions can be modified Most existing PCP constructions can be modified to obtain robust PCPs of Proximityto obtain robust PCPs of Proximity

However, the parameters of such robust-PCPPs do However, the parameters of such robust-PCPPs do not satisfy our needsnot satisfy our needs

So, build robust PCPP from scratchSo, build robust PCPP from scratch

2626

Bird’s eye-view of PCP constructionBird’s eye-view of PCP constructionF m F m

f 1

. . . . . .f 2

PCP Construction: Sequence of function evaluations, fi : F m ! F

Checks performed by verifier• Each function fi: F

m ! F is a low-degree polynomial

• Input Consistency: f1 ¼ input

• Each fi+1 is obtained consistently from fi

e.g.: fi+1(x) = fi (x)¢ fi (x+1)

• final function fr:Fm ! F is identically zero

i.e., fr ´ 0

How to test if a function is a low-degree polynomial ? Input: Evaluation of function f at each point in F m

Need to check if evaluation of f is close to the evaluation of a low-degree polynomial

f r

F m

2727

Low Degree PolynomialsLow Degree Polynomials Main Tool – Low Degree Polynomial over Finite FieldsMain Tool – Low Degree Polynomial over Finite Fields

(Reed-Muller Codes)(Reed-Muller Codes) FF - finite field, - finite field, ff: F: Fmm !! F F, , mm-variate polynomial over -variate polynomial over

FF, , deg(deg(ff)) = maximal degree of monomial in = maximal degree of monomial in ff

l

f : F m ! F [Schwartz-Zippel] [Schwartz-Zippel]

If If f f g g have degree < have degree < dd, , then then

Fact:Fact:

If If deg(deg(f f ) < ) < dd and and ll – line, then – line, then f f restricted to line restricted to line l l is a is a univariate polynomial of univariate polynomial of degree degree < < dd..

Pr[f (x) = g(x)] · djF j

2828

Low Degree Test (LDT)Low Degree Test (LDT)

Robust Soundness of LDT Robust Soundness of LDT [RS ’92, ALMSS ’92][RS ’92, ALMSS ’92]

ff:F:Fmm !! F F isis -far from low degree, then-far from low degree, then

PrPrll[[ff||ll is far from being low-degree ] > is far from being low-degree ] > (()) Amount of Randomness Required:Amount of Randomness Required:

[RS ’92, ALMSS ’92][RS ’92, ALMSS ’92] 2 points – 2 points – 2 log |2 log |F F mm|| [BSVW ’03][BSVW ’03] derandomized set of linesderandomized set of lines ¼¼ log | log |F F mm||

Input: Table of evaluations of f at each point of F m

Output: Is f low-degree?• Choose a random line l• Read f along line l• Check that restriction of f

along l is a univariate low-degree polynomial

l

f : F m ! F

2929

Robust LDTs via BundlingRobust LDTs via Bundling

f1 f2f r

Each LDT performed separatelyEach LDT performed separately Possible to cheat by having just one of Possible to cheat by having just one of ffii not low-degree not low-degree

----- NOT ROBUST----- NOT ROBUST Bundle evaluations of diff. polys. together and perform LDTs in Bundle evaluations of diff. polys. together and perform LDTs in

parallel (bundling)parallel (bundling) PCPP PCPP on query on query xx returns returns (f(f11(x),f(x),f22(x),…, f(x),…, frr(x))(x))

Robust over larger alphabetRobust over larger alphabet FFrr

Can use error-correcting code to make robust over binary Can use error-correcting code to make robust over binary alphabet. alphabet. PCPP PCPP on query on query xx returns returns ECC(fECC(f11(x),f(x),f22(x),…, f(x),…, frr(x))(x))

3030

Building Block - Robust-PCPPBuilding Block - Robust-PCPP

Randomness:Randomness:

# Queries :# Queries :

Robustness ParameterRobustness Parameter Proximity ParameterProximity Parameter

12

logn + O(loglogn)

constant

pn¢poly logn

3131

Applications to CodingApplications to Coding

• Locally Testable Codes

• Relaxed-Locally Decodable Codes

3232

Locally Testable CodesLocally Testable Codes

Lower BoundsLower Bounds [BHR ’03] [BHR ’03] Random LDPC Codes are not LTCsRandom LDPC Codes are not LTCs

LTC ConstructionsLTC Constructions [GS ’02, BSVW ’03][GS ’02, BSVW ’03] This paperThis paper

k ¡ ! k ¢2p

logk

k ¡ ! k ¢2(logk)²

T

constant# queries

w

w – codeword: w – codeword:

Tester T accepts with Tester T accepts with probability 1 probability 1

w - far from codeword:w - far from codeword:

Tester T accepts with low Tester T accepts with low probabilityprobability

Tester

3333

Locally Decodable CodesLocally Decodable Codes

Hadamard – locally decodable, but poor rateHadamard – locally decodable, but poor rate

Upper Bound:Upper Bound: [BIKR ’02][BIKR ’02] n n ·· 2 2O(k)O(k)

Lower Bound:Lower Bound: [KT ’00][KT ’00] n n ¸̧ k k(1)(1)

D

constant# queries

ci th mesg bit?

r

corruption

If less than n bits corrupted,for all message

bits i Pr[Dr (i) = mi ] ¸ 34

3434

RelaxedRelaxed Locally Decodable Codes Locally Decodable Codes New!New!

This paper:This paper: For every For every > 0 > 0, there exist relaxed-, there exist relaxed-LDCs withLDCs with

D

constant# queries

ci th mesg bit?

r

corruption

If less than n bits corrupted,

for “most’’ message bits i

Pr[Dr (i) = mi ] ¸ 34

For remaining bits

Pr[Dr (i) = ?] ¸ 34

k ¡ ! k1+²

3535

Summary of resultsSummary of results

Defined: Robust PCP of proximityDefined: Robust PCP of proximity Strengthened definition of standard PCPsStrengthened definition of standard PCPs

Composition TheoremComposition Theorem simple, modularsimple, modular

Simpler constructions of PCPsSimpler constructions of PCPs Coding applications:Coding applications:

Locally Testable CodesLocally Testable Codes Relaxed Locally Decodable CodesRelaxed Locally Decodable Codes

3636

The End