Tolerant Locally Testable Codes
Atri Rudra
Qualifying Evaluation Project Presentation
Advisor: Venkatesan Guruswami
1
Fake Motivation
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Elvis Presley is alive!
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Check DNA
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Verify this
Too much work
“Spot Check”
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Accept Elvis
Reject Atri
Bruce Campbell ?
2
Outline of the talk
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Real Motivation
Testing Codes
Previous work
Our Contributions
High Level ideas
Some Details
Open problems
3
Error Correcting Codes
C(x)
x
Encoder
y
Tester
Decoder
x
Hopeless
C(x)
x
Give up
4
Property testing
x
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Verify a property
Oracle access to input
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Does x have the property ?
Make few queries
Probabilistic tester
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T
Accepts correct inputs
Rejects very bad inputs (whp)
0/1
5
Codes
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Mapping C : k!n
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Distance d = min u,v2  (C(u),C(v))
k
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(¢,¢) is Hamming Distance
Rate k/n
[n,k,d]
d/2
d/2
d
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Testing Codes
x
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Property x 2? C
Make few queries
Probabilistic Tester
How good is the tester ?
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Accept x 2 C w.p. 1
Reject x far from C w.p. 2/3
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T
Hamming Distance
Local tester
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Constant number of queries
Sub-linear also interesting
0 w.p.
1 2/3
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Locally Testable Codes
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Who Cares ?
Heart of PCPs
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Alternate Characterization of NP
X 2? L
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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
Close
Far
Give up
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Current Local Testers
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Reject if y is far
Accept if y is close
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By defn accepts only y2 C
Against rationale of codes
Close
y
Far
10
Tolerant Local Testers
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Dist(y,C) <= c1d/n
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y
Dist(y,C) > c2d/n
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Accept w.p >= 2/3
Tolerance
Reject w.p. >= 2/3
Soundness
Close
Far
q(n) queries
(c1,c2,q)- testable
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Prev work (0,O(1),O(1))-testable
Perfect completeness
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The Holy Grail
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Constant rate, linear distance
Constant Query Complexity
Not known even for LTCs
Unique decoding radius
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c1=1/2, c2 ¼ 1/2?
d/2
d/2
d
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Contributions
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LTCs ! tolerant LTCs
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Constant rate
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Sub-linear query complexity
[BS04]
Constant # queries
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No generic “complier”
Slightly Sub-constant rate
[BGHSV04]
Constant c1, c2
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More on Contributions
(Constant # queries, Constant Rate)
Near uniform queries
Sub-constant Rate
Partitioned queries
Sub-linear # queries
Goal: Design codes and tolerant testers
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Where are we now ?
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Real Motivation
Testing Codes
Previous work
Our Contributions
High Level ideas
Some Details
Open problems
15
LTC ! tolerant LTC
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Perfect Completeness
Uniform query pattern
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x
c1= O(1/q) by union bound
Almost uniform is
q is not constant ?
T
1
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Local Tester Revisited
x
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Decision procedure is strict
Accept perturbations
There is a problem
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Local View
Locally approx correct )
Global approx correct
Robustness
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[BS04]
T
1
0
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What is next ?
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Constant rate, linear distance
Sub-linear query complexity
Product of Codes
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[BS04]
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Product of Codes
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C [n,k,d]
C2
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Any row 2 C
Any Column 2 C
[n2,k2,d2]
Tester ?
2C
n
C3
n
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Tester for
2
C
row
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pick row or clm
pick j2[n]
Rj2 C ?
Not known to be robust
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Big open question
True for special cases
C is Reed-Solomon
C is C’2
n
C3?
n
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Larger product of Codes
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3
(C )
Similar definition (3D instead of 2D)
Same test
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2? C2 test
Check all n2 pts
N2/3 queries
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N=n3
2
2
2?2CC
2 C2
Robust!
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[BS04]
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Formal definition of Robustness
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v2n
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r random coin
T(v,r)=miny:T(y_r)=1 dist(v,y)
T(v)=Er[T(v,r)]
T is e-robust
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8 v2n, dist(v,C)· e¢T(v)
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3
C
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Tolerant test
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is tolerant LTC
Restriction is close to C2?
Constant rate
N2/3 queries
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Reduce the # queries
Ct (t-Dimension)
N2/t queries
¼? C2
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Tolerance of
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dist(v,C)· 
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tester
n3/3
h
f2 C3 closest to v
¸ 2n/3 choices of h
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3
C
Dist(vh,fh)·  n2
Averaging argument
If not, for ¸ n/3 h, dist(vh,fh) >  n2
) dist(v,f)> n3/3
dist(vh,C2)·? n2
Similar arguments for other planes
v accepted w.p. ¸ 2/3
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So what do we have now ?
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Constant rate, linear distance
Sublinear query complexity
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n # queries
 =2/t
C has no local tester but Ct has one
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What is next ?
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Slightly sub-constant rate, linear distance
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n=k¢ exp(logk) for any >0
Constant query complexity
Based on PCPs
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[BGHSV04]
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PCP of Proximity
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Variant of PCP introduced in [BGHSV04]
CKT-VAL(T)={x:T(x)=1}
Verifier VT such that
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||=s¢ exp(logs)
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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
9 
8
x
s=|T|
Constant # queries
T
VT
1
0
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Local Tester 1.0
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Start with good code C0
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Use PCPP as an aid
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Constant rate and linear distance
Linear size encoding circuit
C1(x)= hC0(x),(x)i
There is a problem
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|x|/|(x)|=o(1)
Distance of C1 is bad
(x)
x
x
(x)
C0
1
0
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Local Tester 1.1
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Increase the “code” part
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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
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Just corrupt the proof part
Corrupted word still close to C2
(C0(x))t
(x)
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Tolerant Local Tester 1.2
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Keep the code and proof parts comparable
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C3(x)=h(C0(x))k,((x))li
k¢|C0(x)|=(l¢|(x)|)
Need near uniform queries
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Constant query complexity
Slightly sub-constant rate, Linear distance
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Used in relaxed LDC in [BGHSV04]
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To summarize
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Defined tolerant LTCs
Explicit constructions
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Constant # queries, slightly sub-constant rate
Sub-linear # queries, constant rate
Both constructions start from some C0
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C0 does not have a (tolerant) local tester
31
Open Questions
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Is “natural” tester for C2 robust ?
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e-robust for e=O(1)
row
No lower bounds on n for LTCs
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Does tolerance make lower bounds easier ?
n
C3?
n
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Questions ?
33