1. September 20, 2010 Invariance in Property Testing 1 Madhu Sudan Microsoft/MIT TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A

2. of 30 Modern challenge to Algorithm Design Data = Massive; Computers = Tiny Data = Massive; Computers = Tiny How can tiny computers analyze massive data? How can tiny computers analyze massive data? Only option: Design sublinear time algorithms. Only option: Design sublinear time algorithms. Algorithms that take less time to analyze data, than it takes to read/write all the data. Algorithms that take less time to analyze data, than it takes to read/write all the data. Can such algorithms exist? Can such algorithms exist? September 20, 2010 Invariance in Property Testing 2

3. of 30 Yes! Polling … Is the majority of the population Red/Blue Is the majority of the population Red/Blue Can find out by random sampling. Can find out by random sampling. Sample size / margin of error Sample size / margin of error Independent of size of population Independent of size of population Other similar examples: (can estimate other moments …) Other similar examples: (can estimate other moments …) September 20, 2010 Invariance in Property Testing 3

4. of 30 Recent “novel” example Can test for homomorphisms: Can test for homomorphisms: Given: f: G  H (G,H finite groups), is f essentially a homomorphism? Given: f: G  H (G,H finite groups), is f essentially a homomorphism? Test: Test: Pick x,y in G uniformly, ind. at random; Pick x,y in G uniformly, ind. at random; Verify f(x) ¢ f(y) = f(x ¢ y) Verify f(x) ¢ f(y) = f(x ¢ y) Completeness: accepts homomorphisms w.p. 1 Completeness: accepts homomorphisms w.p. 1 (Obvious) (Obvious) Soundness: Rejects f w.p prob. Proportional to its “distance” (margin) from homomorphisms. Soundness: Rejects f w.p prob. Proportional to its “distance” (margin) from homomorphisms. (Not obvious) (Not obvious) September 20, 2010 Invariance in Property Testing 4

5. of 30 September 20, 2010 Invariance in Property Testing 5 Brief History [Blum,Luby,Rubinfeld – S’90] [Blum,Luby,Rubinfeld – S’90] Linearity + application to program testing Linearity + application to program testing [Babai,Fortnow,Lund – F’90] [Babai,Fortnow,Lund – F’90] Multilinearity + application to PCPs (MIP). Multilinearity + application to PCPs (MIP). [Rubinfeld+S.] [Rubinfeld+S.] Low-degree testing Low-degree testing [Goldreich,Goldwasser,Ron] [Goldreich,Goldwasser,Ron] Graph property testing Graph property testing Since then … many developments Since then … many developments Graph properties Graph properties Statistical properties Statistical properties … More algebraic properties More algebraic properties

6. of 30 September 20, 2010 Invariance in Property Testing 6 Property Testing Data = a function from D to R: Data = a function from D to R: Property P µ {D  R} Property P µ {D  R} Distance Distance δ(f,g) = Pr x 2 D [f(x) ≠ g(x)] δ(f,g) = Pr x 2 D [f(x) ≠ g(x)] δ(f,P) = min g 2 P [δ(f,g)] δ(f,P) = min g 2 P [δ(f,g)] f is ε-close to g (f ¼ ² g) iff δ(f,g) · ε. f is ε-close to g (f ¼ ² g) iff δ(f,g) · ε. Local testability: Local testability: P is (k, ε, δ)-locally testable if 9 k-query test T P is (k, ε, δ)-locally testable if 9 k-query test T f 2 P ) T f accepts w.p. 1-ε. f 2 P ) T f accepts w.p. 1-ε. δ(f,P) > δ ) T f accepts w.p. ε. δ(f,P) > δ ) T f accepts w.p. ε. Notes: want k(ε, δ) = O(1) for ε,δ= (1). Notes: want k(ε, δ) = O(1) for ε,δ= (1).

7. September 20, 2010 Invariance in Property Testing 7 Why is BLR special?

8. of 30 Why is BLR special? Impressive collection of generalizations, alternate proofs, applications (all of PCP, LTC theory, e.g.)? Impressive collection of generalizations, alternate proofs, applications (all of PCP, LTC theory, e.g.)? Why is it more interesting than just polling? Why is it more interesting than just polling? Why did the proof work? Was it a one-shot thing? Why did the proof work? Was it a one-shot thing? Most previous attempts to extend “broadly” failed … Most previous attempts to extend “broadly” failed … September 20, 2010 Invariance in Property Testing 8

9. of 30 BLR Analysis Fix f s.t. Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Fix f s.t. Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Define g(x) = majority y {Vote x (y)}, Define g(x) = majority y {Vote x (y)}, where Vote x (y) = f(x+y) – f(y). where Vote x (y) = f(x+y) – f(y). Step 0: Show δ(f,g) small Step 0: Show δ(f,g) small Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 2: Use above to show g is well-defined and a homomorphism. Step 2: Use above to show g is well-defined and a homomorphism. September 20, 2010 Invariance in Property Testing 9

10. of 30 Key Step: Step 1 Why is f(x+y) – f(y) = f(x+z) – f(z), usually? Why is f(x+y) – f(y) = f(x+z) – f(z), usually? (Note: Prob over y,z for fixed x.) (Note: Prob over y,z for fixed x.) Proof: Proof: f(x+y) + f(z) = f(x+y+z) [w.h.p.] f(x+y) + f(z) = f(x+y+z) [w.h.p.] = f(x+z) + f(y) [w.h.p. again] = f(x+z) + f(y) [w.h.p. again] Proof from the Book. Proof from the Book. (Indisputable! Inexplicable!) (Indisputable! Inexplicable!) September 20, 2010 Invariance in Property Testing 10

11. of 30 Extensions [Rubinfeld + S. 92-96]: Low degree tests [Rubinfeld + S. 92-96]: Low degree tests [Rubinfeld 94]: Functional equations [Rubinfeld 94]: Functional equations [ALMSS, etc. ]: PCP theory [ALMSS, etc. ]: PCP theory [AKKLR 02]: Reed-Muller tests [AKKLR 02]: Reed-Muller tests [KaufmanRon, JPRZ]: Generalized RM tests. [KaufmanRon, JPRZ]: Generalized RM tests. … each time a new proof of key step. … each time a new proof of key step. September 20, 2010 Invariance in Property Testing 11

12. of 30 Abstraction of BLR (in special case) Restrict to G = F n and H = F Restrict to G = F n and H = F (F = finite field; with q elements) (F = finite field; with q elements) Property: Property: Linear: (sum of linear functions is linear) Linear: (sum of linear functions is linear) Locally characterized: 8 x,y f(x) + f(y) = f(x+y) Locally characterized: 8 x,y f(x) + f(y) = f(x+y) Linear-invariant: Linear function remains linear after linear transformation of domain. Linear-invariant: Linear function remains linear after linear transformation of domain. Single-orbit: Constraints above given by one constraint and implication of linear-invariance. Single-orbit: Constraints above given by one constraint and implication of linear-invariance. Our hope: Such abstractions explain, extend and unify algebraic property testing. Our hope: Such abstractions explain, extend and unify algebraic property testing. September 20, 2010 Invariance in Property Testing 12

13. of 30 Invariances Property P invariant under permutation (function) ¼ : D  D, if Property P invariant under permutation (function) ¼ : D  D, if f 2 P ) f ο ¼ 2 P Property P invariant under group G if Property P invariant under group G if 8 ¼ 2 G, P is invariant under ¼. 8 ¼ 2 G, P is invariant under ¼. Can ask: Does invariance of P w.r.t. “nice” G leads to local testability? Can ask: Does invariance of P w.r.t. “nice” G leads to local testability? September 20, 2010 Invariance in Property Testing 13

14. of 30 Invariances are the key? “Polling” works well when (because) invariant group of property is the full symmetric group. “Polling” works well when (because) invariant group of property is the full symmetric group. Modern property tests work with much smaller group of invariances. Modern property tests work with much smaller group of invariances. Graph property ~ Invariant under vertex renaming. Graph property ~ Invariant under vertex renaming. Algebraic Properties & Invariances? Algebraic Properties & Invariances? September 20, 2010 Invariance in Property Testing 14

15. of 30 Example motivating symmetry Conjecture (AKKLR ‘96): Conjecture (AKKLR ‘96): Suppose property P is a vector space over F 2 ; Suppose property P is a vector space over F 2 ; Suppose its “invariant group” is “2-transitive”. Suppose its “invariant group” is “2-transitive”. Suppose P satisfies a k-ary constraint Suppose P satisfies a k-ary constraint 8 f 2 P, f( ® 1 ) +  + f( ® k ) = 0. 8 f 2 P, f( ® 1 ) +  + f( ® k ) = 0. Then P is (q(k), ² (k,δ),δ)-locally testable. Then P is (q(k), ² (k,δ),δ)-locally testable. Inspired by “low-degree” test over F 2. Implied all previous algebraic tests (at least in weak forms). Inspired by “low-degree” test over F 2. Implied all previous algebraic tests (at least in weak forms). September 20, 2010 Invariance in Property Testing 15

16. of 30 Abstracting Algebraic Properties [Kaufman & S.] [Kaufman & S.] Range is a field F and P is F-linear. Range is a field F and P is F-linear. Domain is a vector space over F (or some field K extending F). Domain is a vector space over F (or some field K extending F). Property is invariant under affine (sometimes only linear) transformations of domain. Property is invariant under affine (sometimes only linear) transformations of domain. “Property characterized by single constraint, and its orbit under affine (or linear) transformations.” “Property characterized by single constraint, and its orbit under affine (or linear) transformations.” September 20, 2010 Invariance in Property Testing 16

17. of 30 Terminology k-Constraint: Sequence of k elements of domain, and set of forbidden values for this sequence. k-Constraint: Sequence of k elements of domain, and set of forbidden values for this sequence. e.g. f(a) + f(b) = f(a+b) e.g. f(a) + f(b) = f(a+b) k-characterization: Collection of k-constraints, satisfaction of which is necessary and sufficient criterion for satisfying property k-characterization: Collection of k-constraints, satisfaction of which is necessary and sufficient criterion for satisfying property e.g. f(a) + f(b) = f(a+b), f(c) + f(d) = f(c+d) … e.g. f(a) + f(b) = f(a+b), f(c) + f(d) = f(c+d) … k-single-orbit characterization: One k-constraint such that its translations under affine group yields k-characterization. k-single-orbit characterization: One k-constraint such that its translations under affine group yields k-characterization. f(L(a)) + f(L(b)) = f(L(a+b)) ; a,b fixed, all linear L. f(L(a)) + f(L(b)) = f(L(a+b)) ; a,b fixed, all linear L. September 20, 2010 Invariance in Property Testing 17

18. September 20, 2010 Invariance in Property Testing 18 Main Results

19. of 30 Some results If P is affine-invariant and has k-single orbit characterization then it is (k, δ/k 3, δ)-locally testable. If P is affine-invariant and has k-single orbit characterization then it is (k, δ/k 3, δ)-locally testable. Unifies previous algebraic tests (in basic form) with single proof. Unifies previous algebraic tests (in basic form) with single proof. September 20, 2010 Invariance in Property Testing 19

20. of 30 Analysis of Invariance-based test Property P given by ® 1,…, ® k ; V µ F k Property P given by ® 1,…, ® k ; V µ F k P = {f | (f(A( ® 1 )), …, f(A( ® k ))) 2 V, P = {f | (f(A( ® 1 )), …, f(A( ® k ))) 2 V, 8 affine A:K n K n } 8 affine A:K n K n } Rej(f) = Prob A [ (f(A( ® 1 )), …, f(A( ® k ))) not in V ] Rej(f) = Prob A [ (f(A( ® 1 )), …, f(A( ® k ))) not in V ] Wish to show: If Rej(f) < 1/k 3, Wish to show: If Rej(f) < 1/k 3, then δ(f,P) = O(Rej(f)). then δ(f,P) = O(Rej(f)). September 20, 2010 Invariance in Property Testing 20

21. of 30 BLR Analog Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Define g(x) = majority y {Vote x (y)}, Define g(x) = majority y {Vote x (y)}, where Vote x (y) = f(x+y) – f(y). where Vote x (y) = f(x+y) – f(y). Step 0: Show δ(f,g) small Step 0: Show δ(f,g) small Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 2: Use above to show g is well-defined and a homomorphism. Step 2: Use above to show g is well-defined and a homomorphism. September 20, 2010 Invariance in Property Testing 21

22. of 30 Generalization g(x) = ¯ that maximizes, over A s.t. A( ® 1 ) = x, g(x) = ¯ that maximizes, over A s.t. A( ® 1 ) = x, Pr A [( ¯,f(A( ® 2 ),…,f(A( ® k ))) 2 V] Pr A [( ¯,f(A( ® 2 ),…,f(A( ® k ))) 2 V] Step 0: δ(f,g) small. Step 0: δ(f,g) small. Vote x (A) = ¯ s.t. ( ¯, f(A( ® 2 ))…f(A( ® k ))) 2 V Vote x (A) = ¯ s.t. ( ¯, f(A( ® 2 ))…f(A( ® k ))) 2 V (if such ¯ exists) (if such ¯ exists) Step 1 (key): 8 x, whp Vote x (A) = Vote x (B). Step 1 (key): 8 x, whp Vote x (A) = Vote x (B). Step 2: Use above to show g 2 P. Step 2: Use above to show g 2 P. September 20, 2010 Invariance in Property Testing 22

23. of 30 BLR Analysis of Step 1 Why is f(x+y) – f(y) = f(x+z) – f(z), usually? Why is f(x+y) – f(y) = f(x+z) – f(z), usually? September 20, 2010 Invariance in Property Testing 23 - f(x+z) f(y) - f(x+y) f(z) -f(y) f(x+y+z) -f(z) 0 ?

24. of 30 Matrix Magic? September 20, 2010 Invariance in Property Testing 24 A( ® 2 ) B( ® k ) B( ® 2 ) A( ® k ) x t Say A( ® 1 ) … A( ® t ) independent; rest dependent t Random No Choice Doesn’t Matter!

25. of 30 Results (contd.) Thm 2: If P is affine-invariant over K and has a single k-local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) Thm 2: If P is affine-invariant over K and has a single k-local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) Proof ingredients: Proof ingredients: Analysis of all affine invariant properties. Analysis of all affine invariant properties. Characterization of all affine invariant properties in terms of degrees of monomials in support of polynomials in family Characterization of all affine invariant properties in terms of degrees of monomials in support of polynomials in family Rough characterization of locality of constraints, in terms of degrees. Rough characterization of locality of constraints, in terms of degrees. Infinitely many (new) properties … Infinitely many (new) properties … September 20, 2010 Invariance in Property Testing 25

26. of 30 Results from [KS ‘08] Thm 1: If P is affine-invariant and has k-single orbit feature then it is (k, δ/k 3, δ)-locally testable. Thm 1: If P is affine-invariant and has k-single orbit feature then it is (k, δ/k 3, δ)-locally testable. Unifies previous algebraic tests with single proof. Unifies previous algebraic tests with single proof. Thm 2: If P is affine-invariant over K and has a single k-local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) Thm 2: If P is affine-invariant over K and has a single k-local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) (explains the AKKLR optimism) (explains the AKKLR optimism) Completely characterizes local testability of affine-invariant properties over vector spaces over small fields. Completely characterizes local testability of affine-invariant properties over vector spaces over small fields. September 20, 2010 Invariance in Property Testing 26

27. of 30 Vector spaces over big fields? Most general case: Most general case: f : K  F m f : K  F m Most interesting cases Most interesting cases K = huge field; F, m small. K = huge field; F, m small. Reasons to study: Reasons to study: Broader class: Potential counterexamples to intuitive beliefs. Broader class: Potential counterexamples to intuitive beliefs. Include starting point for all LTCs (so far). Include starting point for all LTCs (so far). September 20, 2010 Invariance in Property Testing 27

28. of 30 Subsequent results [GrigorescuKaufmanS’08]: Counterexample to AKKLR Conjecture. [GrigorescuKaufmanS’08]: Counterexample to AKKLR Conjecture. [GrigorescuKaufmanS.’09]: Single orbit characterization of some BCH (and other) codes. [GrigorescuKaufmanS.’09]: Single orbit characterization of some BCH (and other) codes. [Ben-SassonS.]: Limitations on rate of (O(1)- locally testable) affine-invariant codes. [Ben-SassonS.]: Limitations on rate of (O(1)- locally testable) affine-invariant codes. [KaufmanWigderson]: LDPC codes with invariance (not affine-invariant) [KaufmanWigderson]: LDPC codes with invariance (not affine-invariant) [BhattacharyyaChenS.Xie,Shapira]: Affine- invariant non-linear properties. [BhattacharyyaChenS.Xie,Shapira]: Affine- invariant non-linear properties. September 20, 2010 Invariance in Property Testing 28

29. of 30 Technical nature of questions Given: k points ® 1, …, ® k from K; Given: k points ® 1, …, ® k from K; and set of positive integers D, and set of positive integers D, When is the k x |D| matrix with When is the k x |D| matrix with columns indexed by [k] and rows by D, with columns indexed by [k] and rows by D, with (i,d)th entry being ® i d, of full column rank? (i,d)th entry being ® i d, of full column rank? Nice connections to symmetric polynomials, and we have new results (we think). Nice connections to symmetric polynomials, and we have new results (we think). September 20, 2010 Invariance in Property Testing 29

30. of 30 Broad directions to consider Is every locally characterized affine-invariant property testable? (likely have a counterexample by now ). Is every locally characterized affine-invariant property testable? (likely have a counterexample by now ). What groups of invariances lead to testability? What groups of invariances lead to testability? Is there a subclass of affine-invariant codes that will lead to linear-rate LTCs? (n o(1) -locally testable with linear rate?) Is there a subclass of affine-invariant codes that will lead to linear-rate LTCs? (n o(1) -locally testable with linear rate?) In general … seek invariances In general … seek invariances September 20, 2010 Invariance in Property Testing 30

31. September 20, 2010 Invariance in Property Testing 31 Thanks

32. of 30 Invariance, Orbits and Testability Single constraint implies many Single constraint implies many One for every permutation ¼ 2 Aut(P): One for every permutation ¼ 2 Aut(P): “Orbit of a constraint C” “Orbit of a constraint C” = {C ο ¼ | ¼ 2 Aut(P)} = {C ο ¼ | ¼ 2 Aut(P)} Extreme case: Extreme case: Property characterized by single constraint + its orbit: “Single orbit feature” Property characterized by single constraint + its orbit: “Single orbit feature” Most algebraic properties have this feature. Most algebraic properties have this feature. W.l.o.g. if domain = vector space over small field. W.l.o.g. if domain = vector space over small field. September 20, 2010 Invariance in Property Testing 32

33. of 30 BLR analysis September 20, 2010 Invariance in Property Testing 33

34. of 30 Motivations for this work What makes linearity testing v. different from testing majority? What makes linearity testing v. different from testing majority? What is the essence of the algebra behind (algebraic) property testing? What is the essence of the algebra behind (algebraic) property testing? Eventually … Eventually … Systematize the search for (starting points of) “locally testable codes”. Systematize the search for (starting points of) “locally testable codes”. September 20, 2010 Invariance in Property Testing 34

35. of 30 Graph Property Testing Initiated by [GoldreichGoldwasserRon] Initiated by [GoldreichGoldwasserRon] Initial examples: Initial examples: Is graph bipartite? Is graph bipartite? Is it 3-colorable? Is it 3-colorable? Is it triangle-free (underlying theorem dates back to 80s)? Is it triangle-free (underlying theorem dates back to 80s)? Many intermediate results Many intermediate results Close ties to Szemeredi’s regularity lemma Close ties to Szemeredi’s regularity lemma Culmination: [AlonFisherNewmanSzegedy]: Culmination: [AlonFisherNewmanSzegedy]: Characterization of all testable properties in terms of regularity. Characterization of all testable properties in terms of regularity. September 20, 2010 Invariance in Property Testing 35

36. of 30 September 20, 2010 Invariance in Property Testing 36 Specific Directions in Algebraic P.T. Fewer results Fewer results More Properties More Properties Low-degree (d < q) functions [RS] Low-degree (d < q) functions [RS] Moderate-degree (q < d < n) functions Moderate-degree (q < d < n) functions q=2: [AKKLR] q=2: [AKKLR] General q: [KR, JPRZ] General q: [KR, JPRZ] Long code/Dictator/Junta testing [BGS,PRS] Long code/Dictator/Junta testing [BGS,PRS] BCH codes (Trace of low-deg. poly.) [KL] BCH codes (Trace of low-deg. poly.) [KL] Better Parameters (motivated by PCPs). Better Parameters (motivated by PCPs). #queries, high-error, amortized query complexity, reduced randomness. #queries, high-error, amortized query complexity, reduced randomness.

37. of 30 My concerns … Relatively few results … Relatively few results … Why can’t we get “rich” class of properties that are all testable? Why can’t we get “rich” class of properties that are all testable? Why are proofs so specific to property being tested? Why are proofs so specific to property being tested? What made Graph Property Testing so well- understood? What made Graph Property Testing so well- understood? What is “novel” about Property Testing, when compared to “polling”? What is “novel” about Property Testing, when compared to “polling”? September 20, 2010 Invariance in Property Testing 37

38. of 30 September 20, 2010 Invariance in Property Testing 38 Contrast w. Combinatorial P.T. Algebraic Property = Code! (usually) Universe Universe: {f:D  R} P Don’t care Must reject Must accept P R is a field F; P is linear!

39. of 30 Basic Implications of Linearity [BHR] If P is linear, then: If P is linear, then: Tester can be made non-adaptive. Tester can be made non-adaptive. Tester makes one-sided error Tester makes one-sided error (f 2 P ) tester always accepts). (f 2 P ) tester always accepts). Motivates: Motivates: Constraints: Constraints: k-query test => constraint of size k: k-query test => constraint of size k: value of f at ® 1,… ® k constrained to lie in subspace. value of f at ® 1,… ® k constrained to lie in subspace. Characterizations: Characterizations: If non-members of P rejected with positive probability, then P characterized by local constraints. If non-members of P rejected with positive probability, then P characterized by local constraints. functions satisfying all constraints are members of P. functions satisfying all constraints are members of P. September 20, 2010 Invariance in Property Testing 39

40. of 30 f = assgm’t to left f = assgm’t to left Right = constraints Right = constraints Characterization of P: Characterization of P: P = {f sat. all constraints} P = {f sat. all constraints} Pictorially September 20, 2010 Invariance in Property Testing 40 1 2 3 D {0000,1100, 0011,1111}

41. of 30 Sufficient conditions? Linearity + k-local characterization Linearity + k-local characterization ) k-local testability? ) k-local testability? [BHR] No! [BHR] No! Elegant use of expansion Elegant use of expansion Rule out obvious test; but also any test … of any “q(k)”-locality Rule out obvious test; but also any test … of any “q(k)”-locality Why is characterization insufficient? Why is characterization insufficient? Lack of symmetry? Lack of symmetry? September 20, 2010 Invariance in Property Testing 41

42. of 30 Example: Degree d polynomials Constraint: When restricted to a small dimensional affine subspace, function is polynomial of degree d (or less). Constraint: When restricted to a small dimensional affine subspace, function is polynomial of degree d (or less). #dimensions · d/(K - 1) #dimensions · d/(K - 1) Characterization: If a function satisfies above for every small dim. subspace, then it is a degree d polynomial. Characterization: If a function satisfies above for every small dim. subspace, then it is a degree d polynomial. Single orbit: Take constraint on any one subspace of dimension d/(K-1); and rotate over all affine transformations. Single orbit: Take constraint on any one subspace of dimension d/(K-1); and rotate over all affine transformations. September 20, 2010 Invariance in Property Testing 42

43. of 30 More details Understanding invariant properties: Understanding invariant properties: Recall: all functions from K n to F are Traces of polynomials Recall: all functions from K n to F are Traces of polynomials ( Trace(x) = x + x p + x p 2 + … + x q/p ( Trace(x) = x + x p + x p 2 + … + x q/p where K = F q and F = F p ) where K = F q and F = F p ) If P contains Tr(3x 5 + 4x 2 + 2); then P contains Tr(4x 2 ) … If P contains Tr(3x 5 + 4x 2 + 2); then P contains Tr(4x 2 ) … So affine invariant properties characterized by degree of monomials in family. So affine invariant properties characterized by degree of monomials in family. Most of the study … relate degrees to upper and lower bounds on locality of constraints. Most of the study … relate degrees to upper and lower bounds on locality of constraints. September 20, 2010 Invariance in Property Testing 43

44. of 30 Some results If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) (explains the AKKLR optimism) (explains the AKKLR optimism) Unfortunately, q depends inherently on K, not just F … giving counterexample to AKKLR conjecture [joint with Grigorescu & Kaufman] Unfortunately, q depends inherently on K, not just F … giving counterexample to AKKLR conjecture [joint with Grigorescu & Kaufman] Linear invariance when P is not F-linear: Linear invariance when P is not F-linear: Abstraction of some aspects of Green’s regularity lemma … [ Bhattacharyya, Chen, S., Xie ] Abstraction of some aspects of Green’s regularity lemma … [ Bhattacharyya, Chen, S., Xie ] Nice results due to [Shapira] Nice results due to [Shapira] September 20, 2010 Invariance in Property Testing 44

45. of 30 More results Invariance of some standard codes Invariance of some standard codes E.g. “dual-BCH”: Have k-single orbit feature! So are “more uniformly” testable. E.g. “dual-BCH”: Have k-single orbit feature! So are “more uniformly” testable. [Grigorescu, Kaufman, S.] [Grigorescu, Kaufman, S.] Side effect: New (essentially tight) relationships between Rej AKKLR (f) and δ(f,Degree-d) over F 2 [with Bhattacharyya, Kopparty, Schoenebeck, Zuckerman] Side effect: New (essentially tight) relationships between Rej AKKLR (f) and δ(f,Degree-d) over F 2 [with Bhattacharyya, Kopparty, Schoenebeck, Zuckerman] September 20, 2010 Invariance in Property Testing 45

46. of 30 More results (contd.) One hope: Could lead to “simple, good locally testable code”? One hope: Could lead to “simple, good locally testable code”? (Sadly, not with affine-inv. [Ben-Sasson, S.]) (Sadly, not with affine-inv. [Ben-Sasson, S.]) Still … other groups could be used? [Kaufman+Wigderson] Still … other groups could be used? [Kaufman+Wigderson] Open: Is every locally characterized affine- invariant property locally testable? Open: Is every locally characterized affine- invariant property locally testable? September 20, 2010 Invariance in Property Testing 46

47. of 30 Conclusions Invariance seems to be a nice perspective on “property testing” … Invariance seems to be a nice perspective on “property testing” … Certainly helps unify many algebraic property tests. Certainly helps unify many algebraic property tests. But should be a general lens in sublinear time algorithmics. But should be a general lens in sublinear time algorithmics. September 20, 2010 Invariance in Property Testing 47