1. 1 Tolerant Locally Testable Codes Atri Rudra Qualifying Evaluation Project Presentation Advisor: Venkatesan Guruswami

2. 2 Fake Motivation Elvis Presley is alive!  Verify this Check DNA  Too much work “Spot Check”  Accept Elvis  Reject Atri  Bruce Campbell ?

3. 3 Outline of the talk Real Motivation Testing Codes Previous work Our Contributions High Level ideas Some Details Open problems

4. 4 Error Correcting Codes x Encoder C(x) Decoder xGive up y x C(x) Tester Hopeless

5. 5 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

6. 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

7. 7 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

8. 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. 9 Another motivation C(x) x y x Give up Far Close

10. 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 Far Close

11. 11 Tolerant Local Testers Dist(y,C) <= c 1 d/n  Accept w.p >= 2/3  Tolerance Dist(y,C) > c 2 d/n  Reject w.p. >= 2/3  Soundness q(n) queries (c 1,c 2,q)- testable  Prev work (0,O(1),O(1))-testable  Perfect completeness y Far Close

12. 12 The Holy Grail Constant rate, linear distance Constant Query Complexity Not known even for LTCs Unique decoding radius  c 1 =1/2, c 2 ¼ 1/2? d/2 d

13. 13 Contributions LTCs ! tolerant LTCs  No generic “complier” Constant rate  Sub-linear query complexity  [BS04] Constant # queries  Slightly Sub-constant rate  [BGHSV04] Constant c 1, c 2

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

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

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

17. 17 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

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

19. 19 Product of Codes C [n,k,d]  C 2  Any row 2 C  Any Column 2 C  [n 2,k 2,d 2 ]  Tester ? n n C3C3 2 C

20. 20 Tester for C 2 pick row or clm pick j2[n] R j 2 C ? Not known to be robust  Big open question  True for special cases  C is Reed-Solomon  C is C’ 2 n n C3?C3? row

21. 21 Larger product of Codes (C 3 ) Similar definition (3D instead of 2D) Same test  2 ? C 2 test  Check all n 2 pts  N 2/3 queries N=n 3  Robust! [BS04] 2 C 2 2 ? C 2

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

23. 23 C 3 is tolerant LTC Tolerant test  Restriction is close to C 2 ? Constant rate N 2/3 queries  Reduce the # queries  C t (t-Dimension)  N 2/t queries ¼ ? C 2

24. 24 Tolerance of C 3 tester dist(v,C)·  n 3 /3  f2 C 3 closest to v ¸ 2n/3 choices of h  Dist(v h,f h )·  n 2  Averaging argument  If not, for ¸ n/3 h, dist(v h,f h ) >  n 2  ) dist(v,f)>  n 3 /3 Similar arguments for other planes v accepted w.p. ¸ 2/3 dist(v h,C 2 )· ?  n 2 h

25. 25 So what do we have now ? Constant rate, linear distance Sublinear query complexity  n  # queries   =2/t C has no local tester but C t has one

26. 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. 27 PCP of Proximity Variant of PCP introduced in [BGHSV04] CKT-VAL(T)={x:T(x)=1} Verifier V T such that  x2 CKT-VAL(T), 9  V T (x,  )=1 wp 1  x far from CKT-VAL(T), 8  V T (x,  )=1 wp <1/2  #queries in hx,  i |  |=s¢ exp(log  s)  s=|T| Constant # queries VTVT x  8 9 1 0 T

28. 28 Local Tester 1.0 Start with good code C 0  Constant rate and linear distance  Linear size encoding circuit Use PCPP as an aid  C 1 (x)= hC 0 (x),  (x)i There is a problem  |x|/|  (x)|=o(1)  Distance of C 1 is bad C0C0 x  x) 1 0 x

29. 29 Local Tester 1.1 Increase the “code” part  C 2 (x)=h (C 0 (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 C 2 (C 0 (x)) t  x)

30. 30 Tolerant Local Tester 1.2 Keep the code and proof parts comparable  C 3 (x)=h(C 0 (x)) k,(  (x)) l i  k¢|C 0 (x)|=  (l¢|  (x)|)  Need near uniform queries Constant query complexity Slightly sub-constant rate, Linear distance Used in relaxed LDC in [BGHSV04]

31. 31 To summarize Defined tolerant LTCs Explicit constructions  Constant # queries, slightly sub-constant rate  Sub-linear # queries, constant rate  Both constructions start from some C 0 C 0 does not have a (tolerant) local tester

32. 32 Open Questions Is “natural” tester for C 2 robust ?  e-robust for e=O(1) No lower bounds on n for LTCs  Does tolerance make lower bounds easier ? n n C3?C3? row

33. 33 Questions ?