A UNIFIED FRAMEWORK FOR TESTING LINEAR-INVARIANT PROPERTIES

ARNAB BHATTACHARYYACSAIL, 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: F2n → {0,1}

Examples of testable properties of Boolean functions: Is function f: F2

n → 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.

(Dense) Graph Properties

Graph properties are invariant with respect to vertex relabelings.

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23

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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: F2

n → {0,1} in property P, then f o L also in P for every linear map L: F2

n → 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]

Subspace Hereditariness

Linear-invariant property P is subspace-hereditary if: for any function f: F2

n → {0,1} satisfying P, restriction of f to any linear subspace of F2

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 F2.

All subspace-hereditary linear-invariant properties are testable

Forbidden Linear System

Given m-by-k matrix M over F2, say subset S of F2

n is M-free if 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

Forbidden “Induced” Linear System Given m-by-k matrix M over F2 and a binary

string in {0,1}k, say function f: F2n → {0,1}

is (M,)-free if there is no x = (x1, …,xk) with each xi in F2

n and Mx = 0, such that: f(xi) = 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 = {(M1,), (M2,),…}, a function f: F2

n → {0,1} is F-free if it is (Mi,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 = {(M1,), (M2,),…}, a function f: F2

n → {0,1} is F-free if it is (Mi,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= {(M1,), (M2,),…} is possibly infinite, each is arbitrary, and each Mi 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= {(M1,), (M2,),…} is possibly infinite, each 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)

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

F2n

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: F2n → {0,1},

there is a subspace H of codimension at most T(such that fH

+g is not -regular for < 2n 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.

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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!