Tolerant Locally Testable Codes

1
Tolerant Locally Testable Codes

• Atri Rudra

Qualifying Evaluation Project Presentation Advisor
Venkatesan Guruswami
2
Fake Motivation

• Elvis Presley is alive!
• Verify this
• Check DNA
• Too much work
• Spot Check
• Accept Elvis
• Reject Atri
• Bruce Campbell ?

3
Outline of the talk

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

4
Error Correcting Codes
C(x)
x
Encoder
y
Decoder
x
Give up
5
Property testing
x

• 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)

T
0/1
6
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
7
Testing Codes
x

• 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
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..

9
Another motivation
C(x)
x
y
x
Close
Far
Give up
10
Current Local Testers

• Reject if y is far
• Accept if y is close
• By defn accepts only y2 C
• Against rationale of codes

y
Close
Far
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Tolerant Local Testers

• Dist(y,C) lt c1d/n
• Accept w.p gt 2/3
• Tolerance
• Dist(y,C) gt c2d/n
• Reject w.p. gt 2/3
• Soundness
• q(n) queries
• (c1,c2,q)- testable
• Prev work (0,O(1),O(1))-testable
• Perfect completeness

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The Holy Grail

• Constant rate, linear distance
• Constant Query Complexity
• Not known even for LTCs
• Unique decoding radius
• c11/2, c2 ¼ 1/2?

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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
Sub-constant Rate
Sub-linear queries
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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 ?

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Local Tester Revisited
x

• Decision procedure is strict
• Accept perturbations
• There is a problem
• Local View
• Locally approx correct ) Global approx correct
• Robustness
• BS04

T
1
0
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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 ?

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Tester for C2
row

• pick row or clm
• pick j2n
• Rj2 C ?
• Not known to be robust
• Big open question
• True for special cases
• C is Reed-Solomon
• C is C2

C3?
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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
• Nn3
• Robust!
• BS04

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Formal definition of Robustness

• v2?n
• r random coin
• ?T(v,r)minyT(y_r)1 dist(v,y)
• ?T(v)Er?T(v,r)
• T is e-robust
• 8 v2?n, 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

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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) gt ? n2
• ) dist(v,f)gt? n3/3
• Similar arguments for other planes
• v accepted w.p. 2/3

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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
• nk exp(log?k) for any ?gt0
• Constant query complexity
• Based on PCPs
• BGHSV04

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PCP of Proximity

• Variant of PCP introduced in BGHSV04
• CKT-VAL(T)xT(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 lt1/2
• queries in hx,?i
• ?s exp(log?s)
• sT
• Constant queries

x
?
8
VT
0
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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

?(x)
x
x
?(x)
C0
1
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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)/(tx)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
• kC0(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 eO(1)
• No lower bounds on n for LTCs
• Does tolerance make lower bounds easier ?

row
C3?
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• Questions ?