some 3cnf properties are hard to test eli ben-sasson harvard & mit prahladh harsha mit sofya...

Some 3CNF Properties are Some 3CNF Properties are Hard to TestHard to Test

Eli Ben-SassonHarvard & MIT

Prahladh HarshaMIT

Sofya RaskhodnikovaMIT

June 10, 2003June 10, 2003 -- STOC '03 ---- STOC '03 -- 22

Property - DefinitionProperty - Definition

Property – Set of StringsProperty – Set of Strings

e.g.:e.g.:

100001111111000011111000110110101100100001

Triangle Free Graphs

1.2. Satisfying assignments of a fixed CNF

Æ (:ei,j Ç :ej,k Ç :ek,i)


Property Testing - DefinitionProperty Testing - Definition

[[RubinfeldRubinfeld SudanSudan 96] [Goldreich Goldwasser 96] [Goldreich Goldwasser Ron 98]Ron 98]

V ClassicalVerifierProbabilis

ticTester


Definition - ContdDefinition - Contd

YESYES: Verifier accepts with : Verifier accepts with probability 1probability 1

FAR from YESFAR from YES: Accepts with : Accepts with low probabilitylow probability FAR – terms of hamming FAR – terms of hamming

distancedistance

ExtensionsExtensions 2 sided Error2 sided Error Adaptive QuestionsAdaptive Questions

YES

FAR fromYES


Property Testing - UsesProperty Testing - Uses

Naturally arise in contexts of PCPsNaturally arise in contexts of PCPs

Massive Data SetsMassive Data Sets Eg: WWW, DNA samples, High Resolution Eg: WWW, DNA samples, High Resolution

ImagesImages Large Access TimeLarge Access Time Need to efficiently check if data satisfies Need to efficiently check if data satisfies

certain propertiescertain properties


Testable PropertiesTestable Properties

Testable with constant number of queriesTestable with constant number of queries bipartiteness [GGR 98]bipartiteness [GGR 98] k-colorability [GGR 98]k-colorability [GGR 98] membership in a regular language [AKNS membership in a regular language [AKNS

99]99] Testing if function is linear [BLR 90]Testing if function is linear [BLR 90]


Non-Testable PropertiesNon-Testable Properties

[GGR 98] prove there exist properties not [GGR 98] prove there exist properties not testable even with a linear number of testable even with a linear number of queries. (Probabilistic Construction)queries. (Probabilistic Construction)

Explicit Linear Lower boundsExplicit Linear Lower bounds 3 Colorable Bounded Degree Graphs [BOT 02]3 Colorable Bounded Degree Graphs [BOT 02] Polynomials of degree n/2 represented as Polynomials of degree n/2 represented as

function evaluation [Sudan]function evaluation [Sudan]


Testable Properties – Local ViewsTestable Properties – Local Views

Property is testable Property is testable )) string far from string far from property has lot of property has lot of local viewslocal views showing showing violation.violation.

Lower Bounds of [BOT 02] and [Sudan] Lower Bounds of [BOT 02] and [Sudan] exploit the fact that there are exploit the fact that there are no small no small local viewslocal views showing violation. showing violation.

traingle free graphs


Properties as CNF formulaeProperties as CNF formulae

Strings (length n)Strings (length n) Has property ?Has property ?

001101…..1001101…..1 XX010101…..0010101…..0 ££

Each property can be represented as a CNF formula.

Triangle Free Graphs Æ (:ei,j Ç :ej,k Ç :ek,i)


CNF Property TestingCNF Property Testing

CNF Property Testing: CNF Property Testing:

For a For a fixed CNFfixed CNF (i.e., property), given an (i.e., property), given an assignment, is itassignment, is it A satisfying assignment? OrA satisfying assignment? Or Far from satisfying?Far from satisfying?

NoteNote: Different from the testing if CNF is : Different from the testing if CNF is satisfiable or far from satisfiable.satisfiable or far from satisfiable.


Bounds for CNF Property TestingBounds for CNF Property Testing

Some CNF properties - hard to test [GGR Some CNF properties - hard to test [GGR 98]98]

2CNF Property Testing: Testable with 2CNF Property Testing: Testable with O(O(ppn)n) queries [FLNRRS 02] queries [FLNRRS 02]

What about kCNFs (k > 2)?What about kCNFs (k > 2)? ““Possibly testable”: there exist “witness” of Possibly testable”: there exist “witness” of

size k that falsifies kCNF.size k that falsifies kCNF.


3CNF Property Testing - Hard3CNF Property Testing - Hard

Main Theorem:Main Theorem:

There exist 3CNF formulae that require There exist 3CNF formulae that require linear numberlinear number of queries, even with of queries, even with adaptive 2-sided error tests.adaptive 2-sided error tests.


kCNF kLINkCNF kLIN

3CNF:3CNF: (x(x11 ÇÇ :: x x22 ÇÇ x x44) ) ÆÆ ( (:: x x22 ÇÇ x x33 ÇÇ x x11) ) (x (x2525 ÇÇ :: x x1010))

++3LIN:3LIN:

(x(x33 ©© x x55 ©© x x11) ) ÆÆ ( x ( x22 ©© x x33 ©© x x11) ) (x (x2323 ©© x x1111))

Advantages: Can use Linear AlgebraAdvantages: Can use Linear Algebra


Linear PropertiesLinear Properties

Defined by linear constraints. Testing Defined by linear constraints. Testing membership in linear space.membership in linear space.

Variables Constraints

V – set of vectors that satisfy allconstraints.

Right degree · k ) V can berepresented by kLIN

x1

x2

x3

xn

X1 © x2 © x4 = 0 (mod 2)


Lower Bound Proof (Linear property)Lower Bound Proof (Linear property)

1.1. For linear property, For linear property, adaptivityadaptivity and and 2-2-sided errorsided error does not help. does not help.

2.2. Prove Prove sufft. propertiessufft. properties for V to be hard for V to be hard for 1-sided non-adaptive tests.for 1-sided non-adaptive tests.

3.3. Prove Prove random linear spacesrandom linear spaces satisfy satisfy above properties.above properties.

4.4. k large in Step 3. Reduce k large in Step 3. Reduce k k !! 3 3..


Adaptivity and 2-sided ErrorAdaptivity and 2-sided Error

Theorem:Theorem:

For testing linear properties, adaptivity For testing linear properties, adaptivity and 2-sided error do not help.and 2-sided error do not help.

Key Idea:Key Idea: Accept only if no constraints are violated.Accept only if no constraints are violated. To check if a linear constraint is satisfied, the To check if a linear constraint is satisfied, the

order of checking the variables is immaterial. order of checking the variables is immaterial.


Lower Bound ProofLower Bound Proof

Want to prove :Want to prove :

88 prob. Tests prob. Tests TT, , 99 string string xx far from having far from having property,property,

Pr[ T accepts xPr[ T accepts x] is high.] is high.

Sufft. to prove :Sufft. to prove : (by Yao’s MinMax Principle)(by Yao’s MinMax Principle)

99 bad distribution bad distribution BB of strings far from of strings far from property, property, 88 deterministic tests deterministic tests TT,,

PrPrx x ÃÃ B B[ T accepts x ][ T accepts x ] is high is high


Bad Distribution - DefintionBad Distribution - Defintion

Distribution B: uniformly pick a basis Distribution B: uniformly pick a basis constraint c, uniformly pick a vector that constraint c, uniformly pick a vector that falsifies only c.falsifies only c.

variables constraints

linearlyindependent constraints

falsifiedconstraint

1

1

1

0


Sufficient Properties – Hard to TestSufficient Properties – Hard to Test

If basis constraints satisfy,If basis constraints satisfy, Property 1: Property 1: -separatedness-separatedness

Property 2: Property 2: (q, (q, )-locality)-locality

then, linear space is hard to test. then, linear space is hard to test.


Property 1:Property 1: -Separatedness-Separatedness

-separated-separated: Any string : Any string xx that falsifies that falsifies exactly exactly oneone basis constraint has basis constraint has large large weightweight. .

falsifiedconstraint

1

1

1

0

w(1110 ) - large

-separatedness

+All strings in B (bad distribution) are far from linear space V.


How can a test detect a string is from the bad How can a test detect a string is from the bad distribution B?distribution B?

(q,(q,) locality) locality: Any : Any dual constraintdual constraint, that is a sum of at , that is a sum of at least least n basis constraints n basis constraints, depends on more , depends on more than than q variablesq variables..

Property 2: (q,Property 2: (q,) – Locality) – Locality

x2 + x3 = 0 (mod 2)

Dual Constraint, which is a sum of large number of basis constraints, depends on few variables.


Probabilistic ConstructionProbabilistic Construction

Properties 1 & 2 are expansion-like Properties 1 & 2 are expansion-like properties. Hence, properties. Hence, random LDPCrandom LDPC codes codes satisfy Properties 1 and 2.satisfy Properties 1 and 2.

However, k (max. right degree) – large.However, k (max. right degree) – large. Reduction Reduction k k !! 3 3::

(x(x11 ©© x x22 ©© x xdd))

++(x(x11 ©© x x22 ©© x xd/2d/2 ©© z) and (x z) and (xd/2+1d/2+1 ©© x xd/2+2d/2+2 ©© x xdd

©© z) z)

This reduction preserves properties 1 and 2. This reduction preserves properties 1 and 2.


Summarizing….Summarizing….

For testing membership in a linear For testing membership in a linear space, adaptivity and 2-sided error do space, adaptivity and 2-sided error do not help.not help.

Random LDPC codes are hard to test Random LDPC codes are hard to test even with a linear number of queries.even with a linear number of queries.

Finally,Finally,

There exist properties describable by There exist properties describable by 3CNFs that are hard to test with linear 3CNFs that are hard to test with linear number of queries, even for adaptive 2-number of queries, even for adaptive 2-sided error tests.sided error tests.

some 3cnf properties are hard to test eli ben-sasson harvard & mit prahladh harsha mit sofya...

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