Tolerant Locally Testable Codes

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Tolerant Locally Testable Codes

Atri Rudra

Qualifying Evaluation Project Presentation

Advisor: Venkatesan Guruswami

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Fake Motivation

Elvis Presley is alive! Verify this

Check DNA Too much work

“Spot Check” Accept Elvis Reject Atri Bruce Campbell ?

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Outline of the talk

Real Motivation Testing Codes Previous work Our Contributions High Level ideas Some Details Open problems

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Error Correcting Codes

x Encoder

C(x)

Decoder

x Give up

y

x C(x)

Tester Hopeless

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Property testing

Verify a property Oracle access to input

Does x have the property ? Make few queries Probabilistic tester

Accepts correct inputs Rejects very bad inputs (whp)

x

T

0/1

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Codes

Mapping C : k!n

Distance d = min u,v2 k (C(u),C(v)) (¢,¢) is Hamming Distance

Rate k/n [n,k,d]

d/2 d/2

d

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Testing Codes

Property x 2? C Make few queries Probabilistic Tester How good is the tester ?

Accept x 2 C w.p. 1 Reject x far from C w.p. 2/3

Hamming Distance

Local tester Constant number of queries Sub-linear also interesting

T

1

x

0 w.p. 2/3

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Locally Testable Codes

Who Cares ? Heart of PCPs

Alternate Characterization of NP X 2? L

Proof (x) Verifier checks (x) Makes q queries

NP = PCP[ O(log n), O(1)] [ALMSS92]…..

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Another motivation

C(x)

x

y

x

Give up

FarClose

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Current Local Testers

Reject if y is far Accept if y is close

By defn accepts only y2 C Against rationale of codes

y

FarClose

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Tolerant Local Testers

Dist(y,C) <= c1d/n Accept w.p >= 2/3 Tolerance

Dist(y,C) > c2d/n Reject w.p. >= 2/3 Soundness

q(n) queries (c1,c2,q)- testable

Prev work (0,O(1),O(1))-testable Perfect completeness

y

FarClose

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The Holy Grail

Constant rate, linear distance Constant Query Complexity Not known even for LTCs Unique decoding radius

c1=1/2, c2 ¼ 1/2?d/2 d/2

d

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Contributions

LTCs ! tolerant LTCs No generic “complier”

Constant rate Sub-linear query complexity [BS04]

Constant # queries Slightly Sub-constant rate [BGHSV04]

Constant c1, c2

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More on Contributions

(Constant # queries, Constant Rate)

Sub-constant Rate Sub-linear # queries

Near uniform queries Partitioned queries

Goal: Design codes and tolerant testers

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Where are we now ?

Real Motivation Testing Codes Previous work Our Contributions High Level ideas Some Details Open problems

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LTC ! tolerant LTC

Perfect Completeness Uniform query pattern

c1= O(1/q) by union bound

Almost uniform is q is not constant ?

x

T

1

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Local Tester Revisited

Decision procedure is strict Accept perturbations There is a problem

Local View Locally approx correct )

Global approx correct Robustness

[BS04] 0

x

T

1

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What is next ?

Constant rate, linear distance Sub-linear query complexity Product of Codes

[BS04]

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Product of Codes

C [n,k,d] C2

Any row 2 C Any Column 2 C [n2,k2,d2]

Tester ? n

n

C3

2 C

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Tester for C2

pick row or clm pick j2[n] Rj2 C ? Not known to be robust

Big open question True for special cases C is Reed-Solomon C is C’2

n

n

C3?

row

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Larger product of Codes (C3)

Similar definition (3D instead of 2D) Same test

2? C2 test Check all n2 pts N2/3 queries

N=n3

Robust! [BS04]

2 C22 C2

2? C2

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Formal definition of Robustness v2n

r random coin T(v,r)=miny:T(y_r)=1 dist(v,y)

T(v)=Er[T(v,r)]

T is e-robust 8 v2n, dist(v,C)· e¢T(v)

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C3 is tolerant LTC

Tolerant test Restriction is close to C2?

Constant rate N2/3 queries

Reduce the # queries Ct (t-Dimension) N2/t queries

¼? C2

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Tolerance of C3 tester

dist(v,C)· n3/3 f2 C3 closest to v

¸ 2n/3 choices of h Dist(vh,fh)· n2

Averaging argument If not, for ¸ n/3 h, dist(vh,fh) > n2

) dist(v,f)>n3/3

Similar arguments for other planes v accepted w.p. ¸ 2/3

dist(vh,C2)·? n2

h

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So what do we have now ?

Constant rate, linear distance Sublinear query complexity

n# queries =2/t

C has no local tester but Ct has one

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What is next ?

Slightly sub-constant rate, linear distance n=k¢ exp(logk) for any >0

Constant query complexity Based on PCPs

[BGHSV04]

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PCP of Proximity

Variant of PCP introduced in [BGHSV04] CKT-VAL(T)={x:T(x)=1} Verifier VT such that

x2 CKT-VAL(T), 9 VT(x,)=1 wp 1 x far from CKT-VAL(T), 8 VT(x,)=1 wp <1/2 #queries in hx,i

||=s¢ exp(logs) s=|T|

Constant # queriesVT

x 89

10

T

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Local Tester 1.0

Start with good code C0

Constant rate and linear distance Linear size encoding circuit

Use PCPP as an aid C1(x)= hC0(x),(x)i

There is a problem |x|/|(x)|=o(1) Distance of C1 is bad

C0

x x)

10

x x)

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Local Tester 1.1

Increase the “code” part C2(x)=h (C0(x))t,(x) i Choose t such that |(x)|/(t¢|x|)=o(1)

Constant query complexity Slightly sub-constant rate, linear distance Not tolerant

Just corrupt the proof part Corrupted word still close to C2

(C0(x))t x)

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Tolerant Local Tester 1.2

Keep the code and proof parts comparable C3(x)=h(C0(x))k,((x))li k¢|C0(x)|=(l¢|(x)|) Need near uniform queries

Constant query complexity Slightly sub-constant rate, Linear distance

Used in relaxed LDC in [BGHSV04]

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To summarize

Defined tolerant LTCs Explicit constructions

Constant # queries, slightly sub-constant rate Sub-linear # queries, constant rate Both constructions start from some C0

C0 does not have a (tolerant) local tester

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Open Questions

Is “natural” tester for C2 robust ? e-robust for e=O(1)

No lower bounds on n for LTCs Does tolerance make lower bounds easier ?

n

n

C3?

row

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Questions ?