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Tolerant Locally Testable Codes

Tolerant Locally Testable Codes. Atri Rudra. Qualifying Evaluation Project Presentation Advisor: Venkatesan Guruswami. Fake Motivation. Elvis Presley is alive! Verify this Check DNA Too much work “Spot Check” Accept Elvis Reject Atri Bruce Campbell ?. Outline of the talk.

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Tolerant Locally Testable Codes

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  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. Tester Hopeless x C(x) 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 0 w.p. 2/3 1

  8. 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

  11. y Close Far 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

  12. d/2 d/2 d The Holy Grail • Constant rate, linear distance • Constant Query Complexity • Not known even for LTCs • Unique decoding radius • c1=1/2, c2¼ 1/2?

  13. Contributions • LTCs ! tolerant LTCs • No generic “complier” • Constant rate • Sub-linear query complexity • [BS04] • Constant # queries • Slightly Sub-constant rate • [BGHSV04] • Constant c1, c2

  14. (Constant # queries, Constant Rate) Near uniform queries Partitioned queries Goal: Design codes and tolerant testers More on Contributions Sub-constant Rate Sub-linear # queries

  15. Where are we now ? • Real Motivation • Testing Codes • Previous work • Our Contributions • High Level ideas • Some Details • Open problems

  16. x T 1 LTC ! tolerant LTC • Perfect Completeness • Uniform query pattern • c1= O(1/q) by union bound • Almost uniform is • q is not constant ?

  17. 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

  18. What is next ? • Constant rate, linear distance • Sub-linear query complexity • Product of Codes • [BS04]

  19. 2 C n C3 n Product of Codes • C [n,k,d] • C2 • Any row 2 C • Any Column 2 C • [n2,k2,d2] • Tester ?

  20. n n Tester for C2 row • 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 C3?

  21. 2 C2 2? C2 2 C2 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]

  22. Formal definition of Robustness • v2n • r random coin • T(v,r)=miny:T(y_r)=1 dist(v,y) • T(v)=Er[T(v,r)] • T is e-robust • 8v2n, dist(v,C)· e¢T(v)

  23. ¼? C2 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

  24. h dist(vh,C2)·?n2 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

  25. 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

  26. What is next ? • Slightly sub-constant rate, linear distance • n=k¢ exp(logk) for any >0 • Constant query complexity • Based on PCPs • [BGHSV04]

  27. 9 T 1 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(logs) • s=|T| • Constant # queries x  8 VT 0

  28. 0 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

  29. 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 (x) (C0(x))t

  30. 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]

  31. 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

  32. n n 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 ? row C3?

  33. Questions ?

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