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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 ?