A UNIFIED FRAMEWORK FOR TESTING LINEAR-INVARIANT PROPERTIES ARNAB BHATTACHARYYA CSAIL, MIT (Joint work with ELENA GRIGORESCU and ASAF SHAPIRA)

Property Testing Does the object have a given property P or is it  -far from having P ? Input Object an  -fraction of the representation of the object needs to be modified Queries P is (one-sided) testable if 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.

Properties of Functions  Origins of property testing in testing algebraic properties 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

Properties of Boolean Functions  For this talk, focus on Boolean functions on the hypercube f: F 2 n → {0,1}  Examples of testable properties of Boolean functions:  Is function f: F 2 n → F 2 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.

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

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]

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 iff it is equal to F - freeness for some collection of graphs F.

Linear Invariance  [Kaufman-Sudan ‘07] observed that most natural properties of Boolean functions invariant under linear transformations of domain  If f: F 2 n → {0,1} in property P, then f o L also in P for every linear map L: F 2 n → F 2 n  [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]

Subspace Hereditariness  Linear-invariant property P is subspace-hereditary if: for any function f: F 2 n → {0,1} satisfying P, restriction of f to any linear subspace of F 2 n also satisfies P.

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

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.

Progress towards conjecture  We show testability of a large subclass of subspace- hereditary properties  Those characterized by forbidding 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 F 2. All subspace-hereditary linear-invariant properties are testable

Forbidden Linear System  Given m-by-k matrix M over F 2, say subset S of F 2 n is M-free if there is no x = (x 1, …,x k ) with each x i 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

Forbidden “Induced” Linear System  Given m-by-k matrix M over F 2 and a binary string  in {0,1} k, say function f: F 2 n → {0,1} is (M,  )-free if there is no x = (x 1, …,x k ) with each x i in F 2 n and Mx = 0, such that: f(x i ) =  i for all i in [k]  Example: With m=1, k=3, M=[1 1 1] and  =001, (M,  )-freeness is property of having no x,y with f(x)=f(y)=0 and f(x+y)=1.

Forbidden Family of Linear Systems  Given fixed collection F = {(M 1,   ), (M 2,   ),…}, a function f: F 2 n → {0,1} is F -free if it is (M i,  i )-free for every i.  Example: If M=[1 1 1],   =111 and   =001 and F ={(M,   ), (M,   )}, then F -freeness is linearity No x, y with f(x) + f(y) + f(x+y) = 1 Similarly for Reed-Muller codes

Forbidden Family of Linear Systems  Given fixed (possibly infinite) collection F = {(M 1,   ), (M 2,   ),…}, a function f: F 2 n → {0,1} is F -free if it is (M i,  i )-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)

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

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

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

Our Main Result  F -freeness is testable, where F = {(M 1,   ), (M 2,   ),…} is possibly infinite, each   is arbitrary, and each M i is of complexity 1.

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

Our Main Result  F -freeness is testable, where F = {(M 1,   ), (M 2,   ),…} is possibly infinite, each   is not necessarily all-ones, and each M i 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)

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

Regularity Partitioning H F2nF2n Restriction not “pseudorandom” Restriction “pseudorandom”  [G ‘05]: Can choose H such that very few shifts are red, and # of cosets independent of n. Say f is “pseudorandom” if it does not correlate well with any nonzero linear function.

Green’s Regularity Lemma For every , given function f: F 2 n → {0,1}, there is a subspace H of codimension at most T(  such that f H +g is not  -regular for <  2 n many shifts g.  -regular: correlation with every nonzero linear function at most .

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

 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. } << One-sided testers and hereditariness Semi-subspace-hereditary property Subspace-hereditary property

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  ) [B.-Xie ’10]  Characterization with respect to other invariance groups?

Thanks!