Property Testing

1. A Unified framework for Testing linear-invariant propertiesarnab bhattacharyyaCSAIL, MIT(Joint work with ELENA GRIGORESCU and ASAF SHAPIRA)

2. Property Testing Does the object have a given property P or is it e-far from having P ? Input Object an e-fraction of the representation of the object needs to be modified Queries P is (one-sided) testableif the number of queries needed to always accept positive inputs and reject negative inputs with probability >90% can be made independent of size of the input.

3. Properties of Functions • Origins of property testing in testing algebraicproperties for program checking & PCP’s[Blum-Luby-Rubinfeld ‘93, Rubinfeld-Sudan ‘96] • Input objects are functions on a vector space • Distance of function to property P measured by smallest Hamming distance to evaluation table of a function satisfying P

4. Properties of Boolean Functions • For this talk, focus on Boolean functions on the hypercube f: F2n → {0,1} • Examples of testable properties of Boolean functions: • Is function f: F2n → F2 linear, i.e. f(x+y)=f(x)+f(y) for all x,y?[BLR ’93] • More generally, is it of degree at most d?[Alon-Kaufman-Krivelevich-Litsyn-Ron ‘03] • Fourier dimensionality and sparsity[Gopalan-O’Donnell-Servedio-Shpilka-Wimmer ‘09] • What are all the testable algebraic properties? Want the “shortest” explanation for testability.

6. (Dense) Graph Properties • Graph properties are invariant with respect to vertex relabelings. 1 • Input graph represented by its adjacency matrix • Distance to property P measured by smallest Hamming distance to adjacency matrix of a graph satisfying P. 3 2 5 4 Examples: bipartiteness,3-colorability, triangle-freeness, … [Goldreich-Goldwasser-Ron ‘98]

7. Testability of Graph Properties • All hereditary graph properties are testable with one-sided error.[Alon-Shapira ‘05] • P is hereditary if for any graph G satisfying P, every induced subgraph of G also satisfies P. • “All” testable properties (with one-sided error) are hereditary! • Full characterization given by[Alon-Fischer-Newman-Shapira ’06], [Borgs-Chayes-Lovasz-Sos-Szegedy-Vesztergombi ’06]

8. Forbidden Induced Subgraphs • Given fixed collection of graphs F, a graph G is said to be F-free if G does not contain any graph in F as an induced subgraph. • Bipartiteness: F is infinite A graph property is hereditary iffit is equal to F-freeness for some collection of graphs F.

10. Linear Invariance • [Kaufman-Sudan ‘07]observed that most natural properties of Boolean functions invariant under linear transformations of domain • If f: F2n → {0,1}in property P, then f o L also in P for every linear map L: F2n → F2n • [KS ‘07]showed testability for linear-invariant properties if they formed a subspace and are “locally characterized” • Challenge to characterize all linear-invariant testable properties [Sudan ‘10]

11. Subspace Hereditariness • Linear-invariant property P is subspace-hereditary if: for any function f: F2n → {0,1} satisfying P, restriction of f to any linear subspace of F2n also satisfies P.

12. Our Main Conjecture All subspace-hereditary linear-invariant properties are testable.

13. Implied Characterization • Implication: A linear-invariant property is one-sided testable “iff” it is subspace-hereditary • Restriction to testers whose behavior doesn’t depend on value of n • “Only if” direction is a theorem [BGS10], not conjecture. Shows importance of notion of subspace-hereditariness.

14. Progress towards conjecture • We show testability of a large subclass of subspace-hereditary properties • Those characterized byforbidding solutions to systems of equations of complexity 1 • Technique: constructing robust arithmetic regularity lemmas • Proof of full conjecture along similar lines would depend on developing arithmetic regularity lemmas with respect to higher-order Gowers norms over F2. All subspace-hereditary linear-invariant properties are testable

15. Forbidden Linear System • Given m-by-k matrix M over F2, say subset S of F2n is M-freeif there is no x = (x1, …,xk) with each xi in S such that Mx = 0. • Example: If M=[1 1 1], then M-freeness is property of having no x, y, x+y all in the set Always a monotone property

16. Forbidden “Induced” Linear System • Given m-by-k matrix M over F2 and a binary string s in {0,1}k, say function f: F2n → {0,1} is (M, s)-freeif there is no x = (x1, …,xk) with each xi in F2n and Mx = 0, such that: • f(xi) = si for all i in [k] • Example: With m=1, k=3, M=[1 1 1] and s=001, (M, s)-freeness is property of having no x,y with f(x)=f(y)=0 and f(x+y)=1.

17. Forbidden Family of Linear Systems • Given fixed collection F = {(M1, s1), (M2, s2),…}, a function f: F2n → {0,1} is F-free if it is (Mi,si)-free for every i. • Example: If M=[1 1 1], s1=111 and s2=001 and F={(M, s1), (M, s2)}, then F-freeness is linearity • No x, y with f(x) + f(y) + f(x+y) = 1 • Similarly for Reed-Muller codes

18. Forbidden Family of Linear Systems • Given fixed (possiblyinfinite) collection F = {(M1, s1), (M2, s2),…}, a function f: F2n → {0,1} is F-free if it is (Mi,si)-free for every i. • Property may no longer be “locally characterized”, a requirement in [Kaufman-Sudan ‘07] • Example: ODD-CYCLE-FREENESS (to be discussed tomorrow by Asaf)

19. Why forbidden linear systems? • Fact: Property P is characterized by F-freeness for some collection Fiff it is a subspace-hereditary linear-invariant property

20. Why forbidden linear systems? • Fact: Property P is characterized by F-freeness iff it is a subspace-hereditary linear-invariant property • Property being subspace-hereditary means certain restrictions to subspaces are forbidden. • Linear systems encode these subspaces, pattern strings encode the forbidden restrictions on them

21. Our Main Conjecture F -freeness is testable, for any fixed collection F.

22. Our Main Result • F-freeness is testable, where F= {(M1, s1), (M2, s2),…} is possibly infinite, each si is arbitrary, and each Mi is of complexity 1.

23. Complexity of Linear Systems • Introduced by[Green-Tao ‘06]. Also called “Cauchy-Schwarz complexity” [Gowers-Wolf ‘07]. • Every system of equations assigned a complexity. Exact definition unimportant for purposes of this talk. • Any system of rank at most 2 is of complexity 1 • Linear systems used to define RM codes of order d have complexity d

24. Our Main Result • F-freeness is testable, where F= {(M1, s1), (M2, s2),…} is possibly infinite, each si is not necessarily all-ones, and each Mi is of complexity 1. • Linearity is testable…once again  • Price of generality: bound on the query complexity is extremely weak in terms of distance parameter (tower of exponentials)

25. Previous Work • Testability results: • [Green ‘05]: (M, s)-freeness for M with rank 1 and s is all-ones. • [B.-Chen-Sudan-Xie ‘09]: (M, s)-freeness for M of complexity 1 and s is all-ones • [Kràl’-Serra-Vena ‘09, Shapira ‘09]: F-freeness where F is finite collection, each M of arbitrary complexity but each s still all-ones

26. Regularity Partitioning • Restriction not “pseudorandom” • Restriction “pseudorandom” • [G ‘05]: Can choose H such that very few shifts are red, and # of cosets independent of n. F2n H Say f is “pseudorandom” if it does not correlate well with any nonzero linear function.

27. Green’s Regularity Lemma For every e, given function f: F2n → {0,1}, there is a subspace H of codimension at mostT(e) such that fH+g is not e-regular for < e2n many shifts g. e-regular: correlation with every nonzero linear function at most e.

28. Regularity Lemma: Functional version Actual statement used in the proof more complicated

29. One-sided testers and hereditariness • A tester T is oblivious if it inspects a uniformly chosen random subspace and then acts the same independent of the value of n • First condition is without loss of generality • Theorem: Any linear-invariant property that is one-sided testable by an oblivious tester is semi-subspace-hereditary. <e } Semi-subspace-hereditary property Subspace-hereditary property

30. Other Open Questions • Testability over other fields? • Testability of non-Boolean functions? • Are there better query complexity upper bounds, even for Green’s problem? • Best lower bound only poly(1/e) [B.-Xie ’10] • Characterization with respect to other invariance groups?

31. Thanks!