Analysis of Boolean Analysis of Boolean FunctionsFunctions

andandComplexity TheoryComplexity Theory

EconomicsEconomicsCombinatoricsCombinatorics

Etc.Etc.

Slides prepared with help of Ricky Slides prepared with help of Ricky RosenRosen

InfluentialInfluential People People The theory of the The theory of the InfluenceInfluence of Variables on of Variables on

Boolean FunctionsBoolean Functions [[KKLKKL,BL,R,M],BL,R,M], has been , has been introduced to tackle introduced to tackle Social ChoiceSocial Choice problems and problems and distributed computingdistributed computing..

It has motivated a magnificent body of It has motivated a magnificent body of work, related towork, related to Sharp Threshold Sharp Threshold [F, FK][F, FK] PercolationPercolation [BKS][BKS] Economics: Economics: Arrow’s TheoremArrow’s Theorem [K][K] Hardness of ApproximationHardness of Approximation [DS][DS]

Utilizing Utilizing Harmonic Analysis of Boolean Harmonic Analysis of Boolean functionsfunctions… …

And the real important question:And the real important question:

Where to go for Dinner?Where to go for Dinner?

The The alternativesalternatives

Diners would cast their vote Diners would cast their vote in an (electronic) envelopein an (electronic) envelope

The system would decide –The system would decide –not necessarily according not necessarily according to majority…to majority…

And what ifAnd what ifsomeonesomeone(in Florida?)(in Florida?)can flipcan flipsome votessome votes

PowerPower

influenceinfluence

0,1f :P[n] 0,1f :P[n]

Boolean FunctionsBoolean Functions

DefDef: : AA Boolean functionBoolean function

[ ] [ ]

1,1

n

P n x n

[ ] [ ]

1,1

n

P n x nPower set

of [n]

1,1 f :P[n] 1,1 f :P[n]

Choose the location of -1

Choose a sequence of -1

and 1

1,4 1,1,1, 1 1,4 1,1,1, 1

Noise SensitivityNoise Sensitivity

The values of the variables may The values of the variables may each, independently, flip with each, independently, flip with probability probability

It turns outIt turns out: one cannot design : one cannot design an an ff that would be robust to that would be robust to such noise --that is, would, on such noise --that is, would, on average, change value w.p. average, change value w.p. < < O(1)O(1)-- unless determining the -- unless determining the outcome according to very few outcome according to very few of the votersof the voters

1-1

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1

DefDef: : thethe influenceinfluence of of ii on on ff is the is the probability, over a random input probability, over a random input xx, that , that ff changes its value when changes its value when ii is flipped is flipped

Voting and Voting and influenceinfluence

ix P n

f Pr f x i f x \ iinfluence

ix P n

f Pr f x i f x \ iinfluence

TheThe influenceinfluence of of ii on on MajorityMajority is the probability, is the probability, over a random input over a random input xx, , MajorityMajority changes with changes with ii

this happens when half of the this happens when half of the n-1n-1 coordinate coordinate (people) vote (people) vote -1-1 and half vote and half vote 11..

i.e. i.e.

MajorityMajority :{1,-1}:{1,-1}nn {{11,,-1-1}}

1 12

1 / 2iinfl uence

nn

n n 1 12

1 / 2iinfl uence

nn

n n

1 ? 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1

ParityParity : : {1,-1}{1,-1}nn {{11,,-1-1}}

n n

i i ji 1 j i

i

Parity(X) x x x

1Influence

n n

i i ji 1 j i

i

Parity(X) x x x

1InfluenceAlways

changes the value of

parity

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1

influence of influence of ii on on DictatorshipDictatorshipii= 1= 1.. influence of influence of jjii on on DictatorshipDictatorshipii== 00..

DictatorshipDictatorshipii :{1,-1}:{1,-1}2020 {{11,,-1-1}} DictatorshipDictatorshipii(x)=x(x)=xii

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1

Average Sensitivity (Total-Average Sensitivity (Total-Influence)Influence)

DefDef: : thethe Average SensitivityAverage Sensitivity of of ff ((asas) ) is the sum of influences of all is the sum of influences of all coordinates coordinates i i [n] [n] ::

asas(Majority) = O(n(Majority) = O(n½½)) asas(Parity) = n(Parity) = n asas(dictatorship) =1(dictatorship) =1

ii

ffas influence ii

ffas influence

When When asas(f)=1(f)=1

DefDef: : ff is a is a balancedbalanced function if it equals function if it equals -1-1 exactly half of the times: exactly half of the times:

EExx[f(x)]=0[f(x)]=0

Can a balanced Can a balanced ff have have asas(f) < 1(f) < 1??

What about What about asas(f)=1(f)=1??

Beside dictatorships?Beside dictatorships?

PropProp: : ff is is balancedbalanced andand asas(f)=1(f)=1 ff is a is a dictatorshipdictatorship..

Representing Representing ff as a as a PolynomialPolynomial What would be the monomials over What would be the monomials over x x

P[n]P[n] ? ?

All powers except All powers except 00 and and 11 cancel out! cancel out!

Hence, one for each Hence, one for each charactercharacter SS[n][n]

These are all the These are all the multiplicative functionsmultiplicative functions

S x

S ii S

(x) x 1

S x

S ii S

(x) x 1

Fourier-Walsh TransformFourier-Walsh Transform

Consider all charactersConsider all characters

Given any functionGiven any functionlet the Fourier-Walsh coefficients of let the Fourier-Walsh coefficients of ff be be

thus thus ff can be described as can be described as

f : P n f : P n

S ii S

(x) x

S ii S

(x) x

S Sx

f S f E f x x S S

xf S f E f x x

SS

ff S SS

ff S

NormsNormsDefDef:: ExpectationExpectation norm on the function norm on the function

DefDef:: SummationSummation norm on the transform norm on the transform

ThmThm [Parseval]: [Parseval]:

HenceHence, for a Boolean , for a Boolean ff

q q

q x P[n]ff (x)

q q

q x P[n]ff (x)

q q

q S n

ff S

q q

q S n

ff S

22

ff 22

ff

2 2

2S

f (S) f 1 2 2

2S

f (S) f 1

SimpleSimple ObservationsObservations

DefDef::

ClaimClaim:For any function :For any function ff whose range whose range is is {-1,0,1}{-1,0,1}::

1 x P[n]

ff (x)

1 x P[n]ff (x)

q 1

q 1 x P[n]ff Pr f(x) { 1,1}

q 1

q 1 x P[n]ff Pr f(x) { 1,1}

Variables` InfluenceVariables` Influence

Recall: Recall: influenceinfluence of an index of an index i i [n][n] on a on a Boolean function Boolean function f:{1,-1}f:{1,-1}nn {1,-1}{1,-1} is is

Which can be expressed in terms of the Which can be expressed in terms of the Fourier coefficients of Fourier coefficients of ff

ClaimClaim::

And the as:And the as:

x P n

(f ) Pr f x f x iiInfluence

x P n

(f ) Pr f x f x iiInfluence

2

S,i S

ff SiInfluence

2

S,i S

ff SiInfluence

2

S

f = f S Sas 2

S

f = f S Sas

Fourier Representation of Fourier Representation of influenceinfluence

ProofProof: consider the influence : consider the influence functionfunction

which in Fourier representation iswhich in Fourier representation is

andand

i

f x f x if x

2

i

f x f x if x

2

i S S SS S

Si S

1 1f x f(S) x f(S) x i

2 2

f(S) x

i S S SS S

Si S

1 1f x f(S) x f(S) x i

2 2

f(S) x

22

i i 2i S

ff x f (S)

influence 22

i i 2i S

ff x f (S)

influence

Balanced Balanced ff s.t. s.t. asas(f)=1(f)=1 is is Dict.Dict.

Since Since ff is balanced and is balanced and

So So ff is linear is linear

For any For any ii s.t. s.t.

f 0 f 0

2 2

S S

ˆ ˆf S S f S S f 1as

2 2

S S

ˆ ˆf S S f S S f 1as

ii

f = fi χ ii

f = fi χ

If s s.t |s|>1and

then as(f)>1 f s 0 f s 0

f {i} 0 f {i} 0

i i

f x f x i 2f {i} 2,2

f { f x or,1 f} 1 xi

i i

f x f x i 2f {i} 2,2

f { f x or,1 f} 1 xi

Only i has changed

Expectation and VarianceExpectation and Variance

ClaimClaim::

Hence, for any Hence, for any ff

x

f E f(x)

xf E f(x)

22

x P n x P n

2 22

2S n,S

ff x E f x

ff f S

V E

22

x P n x P n

2 22

2S n,S

ff x E f x

ff f S

V E

First Passage Percolation First Passage Percolation [BKS][BKS]

Each edge costs a w/probability ½ and b w/probability

½

First Passage PercolationFirst Passage Percolation

Consider the GridConsider the Grid

For each edge For each edge ee of choose of choose independentlyindependently wwee = 1 = 1 or or wwee = 2 = 2, each with probability , each with probability ½½

This induces a shortest-path metric onThis induces a shortest-path metric on

ThmThm : The variance of the shortest path : The variance of the shortest path from the origin to vertex from the origin to vertex vv is bounded is bounded from above by from above by O( |v|/ log |v|) O( |v|/ log |v|) [BKS][BKS]

Proof ideaProof idea: The average sensitivity of : The average sensitivity of shortest-path is bounded by that termshortest-path is bounded by that term

dZdZ

dZdZ

dZdZ

DefDef: A : A graph propertygraph property is a subset of is a subset of graphs invariant under isomorphism.graphs invariant under isomorphism.

DefDef: : a a monotonemonotone graph property is a graph property is a graph property graph property PP s.t. s.t. If If P(G)P(G) then for every super-graph then for every super-graph HH of G of G

(namely, a graph on the same set of (namely, a graph on the same set of vertices, which contains all edges of vertices, which contains all edges of GG) ) P(H)P(H) as well. as well.

PP is in fact a Boolean function: is in fact a Boolean function:P: {-1, 1}P: {-1, 1}VV22{-1, 1}{-1, 1}

Graph propertiesGraph properties

Examples of graph Examples of graph propertiesproperties

GG is connected is connected GG is Hamiltonian is Hamiltonian GG contains a clique of size contains a clique of size tt GG is not planar is not planar The clique number of The clique number of GG is larger than that is larger than that

of its complementof its complement The diameter of The diameter of GG is at most is at most ss ... etc .... etc .

What is the What is the influenceinfluence of different of different ee on on PP??

Erdös–Rényi Erdös–Rényi G(n,p)G(n,p) GraphGraph

TheThe Erdös-RényiErdös-Rényi distribution of distribution of random random graphsgraphs

Put an edge between any two vertices w.p.Put an edge between any two vertices w.p. pp

DefinitionsDefinitions

PP – a graph property – a graph property

pp(P)(P) - the probability that a - the probability that a random graph on random graph on nn vertices with vertices with edge probability edge probability pp satisfies satisfies PP. .

GGG(n,p)G(n,p) - - GG is a random graph is a random graph of of nn vertices and edge vertices and edge probability probability pp..

DefDef: Sharp threshold: Sharp threshold

Sharp threshold in monotone graph Sharp threshold in monotone graph property:property: The transition from a property being The transition from a property being

very unlikely to it being very likely is very unlikely to it being very likely is very swiftvery swift..

G satisfies property P

G Does not satisfiesproperty P

ThmThm: : every monotone graph every monotone graph property has a Sharp Thresholdproperty has a Sharp Threshold [FK][FK] Let Let PP be any monotone property of be any monotone property of

graphs on graphs on nn vertices . vertices .

If If pp(P) > (P) > then then

qq(P) > 1-(P) > 1- for for qq == p + cp + c11log(½log(½)/log)/lognn

Proof ideaProof idea: show : show asasp’p’(P)(P), for , for p’>pp’>p, is , is highhigh

ThmThm [Margulis-Russo]: [Margulis-Russo]:

For monotoneFor monotone ff

qq

d (f )(f )

dq

as q

q

d (f )(f )

dq

as

ProofProof [Margulis-Russo]: [Margulis-Russo]:

i

n nq q q

i qi 1 i 1i

d fff (f )

dq q

influence as i

n nq q q

i qi 1 i 1i

d fff (f )

dq q

influence as

Mechanism Design Mechanism Design ProblemProblem

NN agentsagents, each agent , each agent ii has has privateprivate input input ttiiTT. . All other information isAll other information is publicpublic knowledge.knowledge.

Each agent Each agent ii has a has a valuationvaluation for all items: for all items: Each agent wishes to optimize her own utility.Each agent wishes to optimize her own utility.

ObjectiveObjective: minimize : minimize the objective function, the objective function, the total payment.total payment.

MeansMeans: protocol between agents and : protocol between agents and auctioneerauctioneer..

Vickrey-Clarke-Groves Vickrey-Clarke-Groves (VCG)(VCG)

Sealed bid auctionSealed bid auction A A Truth RevealingTruth Revealing protocol, namely, protocol, namely,

one in which each agent might as one in which each agent might as well reveal her valuation to the well reveal her valuation to the auctioneerauctioneer

Whereby each agent gets the best Whereby each agent gets the best (for her) price she could have bid and (for her) price she could have bid and still win the auction still win the auction

Shortest Path using VGCShortest Path using VGC

Problem definition:Problem definition: Communication networkCommunication network modeled by a directed modeled by a directed

graph graph GG and two vertices source and two vertices source ss and target and target tt.. AgentsAgents = edges in = edges in GG Each agent has a cost for sending a single Each agent has a cost for sending a single

message on her edge denote by message on her edge denote by ttee..

ObjectiveObjective:: find the shortest (cheapest) path find the shortest (cheapest) path from from ss to to tt..

MeansMeans:: protocol between agents and protocol between agents and auctioneer.auctioneer.

VCG for Shortest-PathVCG for Shortest-Path

50$

10$

50$

10$Always in the shortest

path

How much will we overpay?How much will we overpay?

SPSP

Every agent gets an extra $1Every agent gets an extra $1 ThmThm[Mahedia,Saberi,S]: expected extra [Mahedia,Saberi,S]: expected extra

pay is pay is asasSPSP (G) (G)

1$1$

1$1$

1$1$

1$1$

1$1$1$1$

1$1$1$1$

1$1$

2$2$2$2$

2$2$

2$2$2$2$

Learning FunctionsLearning Functions

To To learnlearn a function a function f:Gf:GnnGG is to sample it is to sample it in in poly(n|G|/poly(n|G|/)) points and come up with points and come up with some some f’f’ that differs with that differs with ff on at most an on at most an fraction of the pointsfraction of the points MembershipMembership: choose your points: choose your points RandomRandom: w.h.p. over values for a small : w.h.p. over values for a small randomrandom

set of pointsset of points Applications (when learnable): Applications (when learnable):

bioinformatics, economy, etc.bioinformatics, economy, etc. Also [Akavia, Goldwasser, S.]: Also [Akavia, Goldwasser, S.]:

cryptography -- hardcore predicates via cryptography -- hardcore predicates via list-decodinglist-decoding Learning AlgLearning Alg

ConcentratedConcentrated

DefDef: the : the restrictionrestriction of of ff to to is is

DefDef: : ff is a is a concentrated functionconcentrated function if if >0>0, , of of poly(n/poly(n/) ) size s.t.size s.t.

Thm [Kushilevitz, Mansour]: Thm [Kushilevitz, Mansour]: f:f:{0,1}{0,1}nn{0,1}{0,1} concentrated is concentrated is learnablelearnable

Thm[Akavia, Goldwasser, S.]: over Thm[Akavia, Goldwasser, S.]: over any Abelian group any Abelian group f:Gf:GnnGG

S|

S:S

ff (S)

S|

S:S

ff (S)

2

| 2ff

2

| 2ff

characters

weight

…-5 -3 -1 1 3 5…

JuntasJuntas

A function is a A function is a JJ-junta if its value -junta if its value depends on only depends on only JJ variables. variables.

A Dictatorship is 1-juntaA Dictatorship is 1-junta

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1 -1

JuntasJuntas

A function is a A function is a JJ-junta if its value -junta if its value depends on only depends on only JJ variables. variables.

ThmThm [Fischer, Kindler, Ron, Samo., [Fischer, Kindler, Ron, Samo., S]: Juntas are S]: Juntas are testabletestable

ThmThm [ [Kushilevitz, Mansour; Mossel, Kushilevitz, Mansour; Mossel, OdonelOdonel]: Juntas are]: Juntas are learnable learnable

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1

- Noise sensitivity- Noise sensitivity The noise sensitivity of a function f is the probability The noise sensitivity of a function f is the probability

that f changes its value when flipping a subset of its that f changes its value when flipping a subset of its variables according to the variables according to the pp distribution. distribution.

Choose a subset, I, of variablesEach var is in the set with probability

Choose a subset, I, of variablesEach var is in the set with probability

1-1

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1 -1 -11

Flip each value of the subset, Iwith probability p

Flip each value of the subset, Iwith probability p

What is the new value of f?What is the new value of f?

NS f Pr f x f x \ I z,p n n Ix ,I ,zp p

NS f Pr f x f x \ I z,p n n Ix ,I ,zp p

Noise sensitivity and juntasNoise sensitivity and juntas

Juntas are noise insensitive (stable)Juntas are noise insensitive (stable)

ThmThm [Bourgain; Kindler & S]: [Bourgain; Kindler & S]: Noise insensitive (stable) Boolean functions are Noise insensitive (stable) Boolean functions are JuntasJuntas

Choose a subset (I) of variablesEach var is in the set with probability

Choose a subset (I) of variablesEach var is in the set with probability

1-1

1 1 1 1 1 1 1 1 1 11-1 -1-1-1-1-1-1-1-1 -1 -11

Flip each value of the subset (I)with probability p

Flip each value of the subset (I)with probability p

What is the new value of f?W.H.P STAY THE SAME

What is the new value of f?W.H.P STAY THE SAME

JuntaJunta

Freidgut TheoremFreidgut Theorem

ThmThm: any Boolean : any Boolean ff is an is an [[, j]-, j]-junta for junta for

ProofProof::1.1. Specify the junta Specify the junta JJ

2.2. Show the complement ofShow the complement of JJ has little influence has little influence

f /O asj = 2 f /O asj = 2

Long-CodeLong-Code

In the long-code the set of legal-words consists of all In the long-code the set of legal-words consists of all monotone dictatorshipsmonotone dictatorships

This is the most extensive binary code, as its bits This is the most extensive binary code, as its bits represent all possible binary values over represent all possible binary values over nn elements elements

Long-CodeLong-Code

Encoding an element Encoding an element ee[n][n] :: EEee legally-encodeslegally-encodes an element an element ee if if EEee = f = fee

FF FF TT TT TT

Open QuestionsOpen Questions

Hardness of ApproximationHardness of Approximation:: MAX-CUTMAX-CUT ColoringColoring a 3-colorable graph with fewest colors a 3-colorable graph with fewest colors

Graph PropertiesGraph Properties: find sharp-thresholds for : find sharp-thresholds for propertiesproperties

Circuit ComplexityCircuit Complexity: switching lemmas: switching lemmas Mechanism DesignMechanism Design: show a non truth-revealing : show a non truth-revealing

protocol in which the pay is smaller (Nash protocol in which the pay is smaller (Nash equilibrium when all agents tell the truth?)equilibrium when all agents tell the truth?)

AnalysisAnalysis: show weakest condition for a function to : show weakest condition for a function to be a Juntabe a Junta

LearningLearning: by random queries: by random queries Apply Apply Concentration of MeasureConcentration of Measure techniques to techniques to

other problems in Complexity Theoryother problems in Complexity Theory

Where to go for Dinner?Where to go for Dinner?

The The alternativesalternatives

Diners would cast their vote Diners would cast their vote in an (electronic) envelopein an (electronic) envelope

The system would decide –The system would decide –not necessarily according not necessarily according to majority…to majority…

And what ifAnd what ifsomeonesomeone(in Florida?)(in Florida?)can flipcan flipsome votessome votes

PowerPower

influenceinfluence

Of course they’ll have to discuss it

over dinner….

Low-degree B.f are JuntasLow-degree B.f are Juntas

CorollaryCorollary: :

fix a fix a pp-biased distribution -biased distribution pp over over

P([n])P([n])

Let Let >0>0 be any parameter. be any parameter.

Set Set k=logk=log1-1-(½)(½)

Then Then constant constant >0>0 s.t. any Boolean s.t. any Boolean

function function f:P([n])f:P([n]){-1,1}{-1,1} satisfying satisfying

is an is an [[,j]-junta ,j]-junta for for j=O(j=O(-2-2kk332k2k))

2k22

f Ok

2k22

f Ok

2ns f O k 2ns f O k

DefDef((,p,x,p,x[n] [n] ): Let ): Let 0<0<<1<1, and , and xxP([n])P([n])

Then Then y~y~,p,x,p,x, if , if y = (x\I)y = (x\I) z z where where I~I~[n][n] is a is a noise subsetnoise subset, and, and z~ z~ pp

II is a is a replacementreplacement..

DefDef((--noise-sensitivitynoise-sensitivity): let ): let 0<0<<1<1, then, then

[ When [ When p=½p=½ equivalent to flipping each equivalent to flipping each coordinate in coordinate in xx independently w.p. independently w.p. /2/2.].]

[n] [n]p ,p,xx~ ,y~

ns f = Pr f x f y

[n] [n]p ,p,xx~ ,y~

ns f = Pr f x f y

[n][n]xIIz

Noise-SensitivityNoise-Sensitivity

Noise-Sensitivity – Cont.Noise-Sensitivity – Cont.

AdvantageAdvantage: very efficiently testable (using : very efficiently testable (using only two queries) by a only two queries) by a perturbation-testperturbation-test..

DefDef ((perturbation-testperturbation-test): choose ): choose x~x~pp, and , and y~y~,p,x,p,x, check whether , check whether f(x)=f(y)f(x)=f(y) The success is proportional to the noise-The success is proportional to the noise-sensitivity of sensitivity of ff..

PropProp: the : the -noise-sensitivity is given by -noise-sensitivity is given by

2S

S

2 ns f =1 1 f S 2S

S

2 ns f =1 1 f S

Relation between Relation between ParametersParameters

PropProp: small : small nsns small small high-freq weighthigh-freq weight

ProofProof::therefore: therefore: if if nsns is small, then is small, then Hence the Hence the high frequencieshigh frequencies must must have small weights (ashave small weights (as ). ).

PropProp: small : small asas small small high-freq weighthigh-freq weight

ProofProof:: 2

S

ff (S) S as 2

S

ff (S) S as

2S

S

2 ns f =1 1 f S 2S

S

2 ns f =1 1 f S

2S

S

1 f S ~1 2S

S

1 f S ~1

2

S

f S 1 2

S

f S 1

High vs. Low FrequenciesHigh vs. Low Frequencies

DefDef: The section of a function : The section of a function ff above above kk is is

and the and the low-frequency low-frequency portion isportion is

kS

S k

ff S

k

SS k

ff S

kS

S k

ff S

k

SS k

ff S

Low-degree B.f are JuntasLow-degree B.f are Juntas

Theorem Theorem [Bourgain, Kindler & S][Bourgain, Kindler & S] : :

constant constant >0>0 s.t. any Boolean function s.t. any Boolean function

f:P([n])f:P([n]){-1,1}{-1,1} satisfying satisfying

is an is an [[,j]-junta ,j]-junta for for j=O(j=O(-2-2kk332k2k))

CorollaryCorollary: :

fix a fix a pp-biased distribution -biased distribution pp over over P([n])P([n])

Let Let >0>0 be any parameter. be any parameter.

Set Set k=logk=log1-1-(½)(½)

Then Then constant constant >0>0 s.t. any Boolean function s.t. any Boolean function

f:P([n])f:P([n]){-1,1}{-1,1} satisfying satisfying

is an is an [[,j]-junta ,j]-junta for for j=O(j=O(-2-2kk332k2k))

2k22

f Ok

2k22

f Ok

2ns f O k 2ns f O k

Specify the JuntaSpecify the Junta

Set Set k=k=(as(f)/(as(f)/),), and and =2=2--(k)(k)

Let Let

We’ll prove:We’ll prove:

and letand let

hence, hence, J J is a is a [[,j]-,j]-junta, and junta, and |J|=2|J|=2O(k)O(k)

iJ i | finfluence iJ i | finfluence

2

J 2A f 1 2

2

J 2A f 1 2

Jf ' (x) sign A f x J Jf ' (x) sign A f x J

Hadamard CodeHadamard Code

In the Hadamard code theIn the Hadamard code theset of legal-words consists of set of legal-words consists of all multiplicative (linear if all multiplicative (linear if over over {0,1}{0,1}) functions) functions

C={C={SS | S | S [n]} [n]}

namely all characters namely all characters

7676

Hadamard Test – SoundnessHadamard Test – Soundness

PropProp(soundness):(soundness):

ProofProof::

1 2 3

1 2 3

1 2 3 3

1 2 3

1 3 2 3

1 2 3

x,y

1 2 3 x,y S S SS ,S ,S

1 2 3 x,y S S S SS ,S ,S

1 2 3 x S S y S SS ,S ,S

3

S

<E [f (x) f (y) f(xy)]=

= f S f S f S E [ (x) (y) (xy)]=

= f S f S f S E [ (x) (y) (x) (y)]=

= f S f S f S E [ (x) (x)] E [ (y) (y)]=

= f S

1+Pr[f (x) f (y) f(xy)]> S [n],f S

2

Testing Long-codeTesting Long-code

DefDef(a (a long-code list-testlong-code list-test): given a code-word ): given a code-word ff, , probe it in a constant number of entries, andprobe it in a constant number of entries, and accept almost always if accept almost always if f f is a monotone is a monotone

dictatorshipdictatorship reject w.h.p if reject w.h.p if ff does not havedoes not have a sizeable fraction a sizeable fraction

of its Fourier weight concentrated on a small set of its Fourier weight concentrated on a small set of variables, that is, if of variables, that is, if a a semi-Juntasemi-Junta JJ[n][n] s.t. s.t.

NoteNote: a long-code list-test, distinguishes : a long-code list-test, distinguishes between the case between the case ff is a is a dictatorshipdictatorship, to the , to the case case ff is far from a is far from a juntajunta..

2

S J

f S

2

S J

f S

Motivation – Testing Long-codeMotivation – Testing Long-code

TheThe long-code list-test long-code list-test are essential tools are essential tools in proving hardness results. in proving hardness results.

Hence finding simple sufficient-conditions Hence finding simple sufficient-conditions for a function to be a junta is important.for a function to be a junta is important.

High Frequencies Contribute High Frequencies Contribute LittleLittlePropProp: : k >> r log rk >> r log r implies implies

ProofProof: a character : a character SS of size larger than of size larger than kk spreads w.h.p. over all parts spreads w.h.p. over all parts IIhh, hence , hence contributes to the influence of all parts.contributes to the influence of all parts.If such characters were heavy If such characters were heavy (>(>/4/4), ), then surely there would be more than then surely there would be more than j j parts parts IIhh that fail the that fail the t t independence-testsindependence-tests

22k

2S k

ff S 4

22k

2S k

ff S 4

AltogetherAltogetherLemmaLemma: :

ProofProof::

Jf 2influence

Jf 2influence

2k k

J J2ff f 2influence + influence

2k kJ J2

ff f 2influence + influence

AltogetherAltogether

k kJ

i J

2

Si S,S ki J 2

ff

f(S) ?

iinfluence influence

k kJ

i J

2

Si S,S ki J 2

ff

f(S) ?

iinfluence influence

Beckner/Nelson/Bonami Beckner/Nelson/Bonami InequalityInequality

DefDef: let : let TT be the following operator on any be the following operator on any ff, ,

PropProp::

ProofProof::

1 / 2z

f x f x zET

1 / 2z

f x f x zET

SS

S n

ff ST

SS

S n

ff ST

S S

S n z

f x f S x zET

S S

S n z

f x f S x zET

Beckner/Nelson/Bonami Beckner/Nelson/Bonami InequalityInequality

DefDef: let : let TT be the following operator on any be the following operator on any ff, ,

ThmThm: for any : for any p≥rp≥r andand ≤((r-1)/(p-1))≤((r-1)/(p-1))½½

1 / 2z

f x f x zET

1 / 2z

f x f x zET

rpffT rpffT

Beckner/Nelson/Bonami Beckner/Nelson/Bonami CorollaryCorollary

Corollary 1Corollary 1: for any real : for any real ff and and 2≥r≥12≥r≥1

Corollary 2Corollary 2: for real : for real f f andand r>2r>2

k

2r2

r 1 fkf k

2r2

r 1 fkf

k

22r

r 1 fkf k

22r

r 1 fkf

Perturbation Perturbation

DefDef: denote by : denote by the distribution the distribution over all subsets of over all subsets of [n][n], which , which assigns probability to a subset assigns probability to a subset xx as follows:as follows:

independently, for each independently, for each ii[n][n], let, let iixx with probability with probability 1-1- iixx with probability with probability

Long-Code TestLong-Code Test

Given a Boolean Given a Boolean ff, choose , choose random random xx and and yy, and choose , and choose zz; check that; check that

f(x)f(y)=f(xyz)f(x)f(y)=f(xyz)

PropProp(completeness): a legal (completeness): a legal long-code word (a long-code word (a dictatorship) passes this test dictatorship) passes this test w.p. w.p. 1-1-

Long-code Tests Long-code Tests

Def Def (a (a long-code testlong-code test): given a code-): given a code-word word ww, probe it in a constant , probe it in a constant number of entries, andnumber of entries, and accept w.h.p if accept w.h.p if ww is a monotone is a monotone

dictatorshipdictatorship reject w.h.p if reject w.h.p if ww is not close to any is not close to any

monotone dictatorshipmonotone dictatorship

Efficient Long-code TestsEfficient Long-code Tests

For some applications, it suffices if the test For some applications, it suffices if the test may accept illegal code-words, nevertheless, may accept illegal code-words, nevertheless, ones which have short list-decoding:ones which have short list-decoding:

DefDef(a (a long-code list-testlong-code list-test): given a code-word ): given a code-word ww, probe it in 2/3 places, and, probe it in 2/3 places, and accept w.h.p if accept w.h.p if w w is a monotone dictatorship,is a monotone dictatorship, reject w.h.p if reject w.h.p if ww is not evenis not even approximately approximately

determined by a short list of domain elementsdetermined by a short list of domain elements, , that is, if that is, if a a JuntaJunta JJ[n][n] s.t. s.t. f f is close to is close to f’ f’ and and f’(x)=f’(xf’(x)=f’(xJ) J) for allfor all x x

NoteNote: a long-code list-test, distinguishes : a long-code list-test, distinguishes between the case between the case ww is a is a dictatorshipdictatorship, to the , to the case case ww is far from a is far from a juntajunta..

Codes and Boolean Codes and Boolean FunctionsFunctions

DefDef: an : an mm-bit code is a subset of the set of -bit code is a subset of the set of all the all the mm-binary string -binary string

CC{-1,1}{-1,1}mm

The The distancedistance of a code of a code CC is the minimum, is the minimum, over all pairs of legal-words (in over all pairs of legal-words (in CC), of the ), of the Hamming distance between the two wordsHamming distance between the two words

NoteNote: A Boolean function over : A Boolean function over nn binary binary variables is a variables is a 22nn-bit string-bit string

Hence, a set of Boolean functions can be Hence, a set of Boolean functions can be considered as a considered as a 22nn-bits code-bits code

Long-Code Long-Code Monotone- Monotone-DictatorshipDictatorship

In the long-code, the legal code-In the long-code, the legal code-words are all monotone dictatorshipswords are all monotone dictatorships

C={C={{i}{i} | i | i [n]} [n]}

namely, all the singleton charactersnamely, all the singleton characters