Analysis of Boolean Functions and Complexity Theory Economics Combinatorics …

Analysis of Boolean Analysis of Boolean FunctionsFunctions

andandComplexity TheoryComplexity Theory

EconomicsEconomicsCombinatoricsCombinatorics

……

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

Boolean FunctionsBoolean Functions [KKL,BL,R,M][KKL,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, FG][F, FG] 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 by not necessarily by majority…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

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

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

21 / 2 12n

n

n

n

iinfl uence

1

21 / 2 12n

n

n

n

iinfl uence

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

ParityParity : : {1,-1}{1,-1}2020 {{11,,-1-1}}

1

Parity( )

1

n n

i i ji j i

i

X x x x

InfluenceAlways

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 i on Dictatorshipinfluence of i on Dictatorshipii= 1.= 1. influence of jinfluence of ji on Dictatorshipi on Dictatorshipii= 0.= 0.

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

Variables` InfluenceVariables` Influence

The influence of a coordinate i The influence of a coordinate i [n] on a [n] on a Boolean function f:{1,-1}Boolean function f:{1,-1}nn {1,-1} is{1,-1} is

The influence of i on f is the probability, The influence of i on f is the probability, over a random input x, that f changes its over a random input x, that f changes its value when i is flipped.value when i is flipped.

i Pr f(x) f(x i )Influence


Average SensitivityAverage Sensitivity of fof f (AS) - The sum (AS) - The sum of influences of all coordinates i of influences of all coordinates i [n]. [n].

# ( ) ( )if x f x i

Average SensitivityAverage Sensitivity of fof f is theis the expectedexpected number of coordinates, for a random number of coordinates, for a random input x, flipping of which changes the input x, flipping of which changes the value of f. value of f.

exampleexample

majority for majority for

What is Average Sensitivity ?What is Average Sensitivity ? AS= ½+ ½+ ½= 1.5AS= ½+ ½+ ½= 1.5

3:{ 1,1} { 1,1}f

1

Influence 2

Influence 3

Influence

Representing f as a Representing f 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 Sx

f S f E f x x

S

S

ff S S

S

ff S

NormsNorms

DefDef:: ExpectationExpectation norm on the function norm on the function

DefDef:: SummationSummation Norm on its Fourier Norm on its Fourier transformtransform

1qq

q x P[n]ff (x)

1qq

q x P[n]ff (x)

1qq

q S [n]

ff (x)

1qq

q S [n]

ff (x)

Fourier Transform: NormFourier Transform: Norm

NormNorm: (: (SumSum))

ThmThm [Parseval]: [Parseval]:

HenceHence, for a Boolean , for a Boolean ff

q q

q S n

ff S

q q

q S n

ff S

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

22

ff 22

ff

2 2

2S

f (S) f 1 2 2

2S

f (S) f 1

1x1x

1 2 nx x ...x1 2 nx x ...x

2x2x

We may think of the Transform as We may think of the Transform as defining a distribution over the defining a distribution over the characters.characters.

2

S

f (S) 1 2

S

f (S) 1

2

S

f (S) 1 2

S

f (S) 1

SimpleSimple ObservationsObservations

Claim:Claim:

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

1 [ ]( )

x P nf f x

1

1 [ ]Pr ( ) { 1,1}

p

p x P nf f f x


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

Which can be expressed in terms of the Which can be expressed in terms of the Fourier coefficients of fFourier coefficients of f

ClaimClaim::

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

Average SensitivityAverage Sensitivity

DefDef: the: the sensitivitysensitivity of x w.r.t. f isof x w.r.t. f is

Thinking of the discrete n-dimensional Thinking of the discrete n-dimensional cube, color each vertex n in color 1 or cube, color each vertex n in color 1 or color -1 (color f(n)).color -1 (color f(n)).

Edge whose vertices are colored with Edge whose vertices are colored with the same color is called monotone.the same color is called monotone.

TheThe average sensitivityaverage sensitivity is the number of is the number of edges whom are not monotone..edges whom are not monotone..

i

# f x f x i i

# f x f x i

average sensitivityaverage sensitivity of of MajorityMajority is the is the expected number of coordinates, for a expected number of coordinates, for a random input x, flipping of which changes random input x, flipping of which changes the value of the value of MajorityMajority. .

Majority Majority :{1,-1}:{1,-1}1919 {{11,,-1-1}}

n i Majority nAS1infl ue ( )ncn

en

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

ii

AS(Majority) I nfl uence(Majority) ii

AS(Majority) I nfl uence(Majority)

Parity Parity :{1,-1}:{1,-1}2020 {{11,,-1-1}}

1

Parity( )n n

i i ji j i

X x x x

Always changes

the value of parity

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

Parity iinfl uence 1 AS( ) n

ii

AS(Parity) I nfl uence(Parity) ii

AS(Parity) I nfl uence(Parity)

influence of i on Dictatorshipinfluence of i on Dictatorshipii= 1.= 1. influence of jinfluence of ji on Dictatorshipi on Dictatorshipii= 0.= 0.

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

iAS(Dictatorship ) =1

i i ii

AS(Dictatorship) = I nfl uence(Dictatorship)i i ii

AS(Dictatorship) = I nfl uence(Dictatorship)

Average SensitivityAverage Sensitivity ClaimClaim::

Proof:Proof:

ˆ 2

s

as f = f s s

ˆ

ˆ

2

i S|i S

2

S

as f = f S

= f S S

When AS(f)=1When AS(f)=1

DefDef: f is a: f is a balancedbalanced function iffunction if THMTHM: f is: f is balancedbalanced

and and as(f)=1as(f)=1 ff is is dictatorshipdictatorship..

ProofProof: : x, sens(x)=1, and as(f)=1 follows.x, sens(x)=1, and as(f)=1 follows. ff is balanced since the dictator is is balanced since the dictator is 11 on on

half of the half of the xx and and -1-1 on half of the on half of the xx..

because only x can change the value of f

xE f(x) 0 xE f(x) 0

When AS(f)=1When AS(f)=1

So f is linearSo f is linear

For i whose For i whose

f 0 f 0

f is balanced

ˆ ˆ2 2

S S

1=as(f ) = f (S) S = f (S) S

ˆ i

i

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

First Passage PercolationFirst Passage Percolation


Choose each edge with probability ½ to be a and ½ to be b


Consider the Grid Consider the Grid

For each edge e of chooseFor each edge e of choose independentlyindependently wwee = a or w = a or wee = b, each with probability ½ 0< a < b = b, each with probability ½ 0< a < b < < . .

This induces a random metric on the vertices ofThis induces a random metric on the vertices of

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

dZ

dZ

dZ


Choose each edge with probability ½ to be 1 and ½ to be 2


Consider the Grid Consider the Grid

For each edge e of chooseFor each edge e of choose independentlyindependently wwee = = 11 or w or wee = = 22, each with probability ½. , each with probability ½.

This induces a random metric on the vertices ofThis induces a random metric on the vertices of

Proposition : The variance of the shortest path Proposition : The variance of the shortest path from the origin to vertex v is bounded by O( |v| from the origin to vertex v is bounded by O( |v| /log |v|). /log |v|).

dZ

dZ

dZ

LetLet G G denote the griddenote the grid

SPSPGG – the shortest path in G from the origin to – the shortest path in G from the origin to v.v.

Let denote the Grid which differ from G Let denote the Grid which differ from G only on wonly on wee i.e. flip coordinate e in G. i.e. flip coordinate e in G.

Set Set

dZ

Proof outlineProof outline

2dSP:{1,2}

.( ) ( ) ( )i isp G SP G SP G

iG

ObservationObservation

e eG

i G

I nfl uence Pr SP(G) SP(σ G)

=Ε ρsp(G)

=pr[e participates in

all the SP

in G]If e participates in

a shortest path then flipping its

value will increase or

decrease the SP in 1 ,if e is not in SP - the SP will

not change.

.( ) ( ) ( )i isp G SP G SP G

Proof cont.Proof cont.

And by [KKL] there is at least one variable And by [KKL] there is at least one variable whose influence was as big as whose influence was as big as (n/logn) (n/logn)

eeG

ee

2

S

2

S

as SP E # SP G SP G

SP

f S S

f S S var SP

Influence

eeG

ee

2

S

2

S

as SP E # SP G SP G

SP

f S S

f S S var SP

Influence

2

S

vvar SP f S S

log v

Graph propertyGraph property

Every Monotone Graph Every Monotone Graph Property has a sharp Property has a sharp

thresholdthreshold

A graph property is a property of A graph property is a property of graphs which is closed under graphs which is closed under isomorphism.isomorphism.

monotone graph property :monotone graph property : Let P be a graph property.Let P be a graph property. Every graph H on the same set of vertices, Every graph H on the same set of vertices,

which contains G as a sub graph satisfies P which contains G as a sub graph satisfies P as well.as well.


Examples of graph Examples of graph propertiesproperties

G is connectedG is connected G is HamiltonianG is Hamiltonian G contains a clique of size tG contains a clique of size t G is not planarG is not planar The clique number of G is larger than The clique number of G is larger than

that of its complementthat of its complement the diameter of G is at most sthe diameter of G is at most s ... etc .... etc .

Erdös – Rényi GraphErdös – Rényi Graph

ModelModel Erdös - Rényi Erdös - Rényi for for random graphrandom graph Choose every edge with probability pChoose every edge with probability p


Model Erdös - Rényi for random graphModel Erdös - Rényi for random graph Choose every edge with probability pChoose every edge with probability p

Every Monotone Graph Every Monotone Graph Property has a sharp Property has a sharp thresholdthreshold

Ehud Friedgut & Gil KalaiEhud Friedgut & Gil Kalai

DefinitionsDefinitions

GNPGNP – a graph property – a graph property

((PP)) - the probability that a random - the probability that a random graph on n vertices with edge graph on n vertices with edge probability p satisfies GP. probability p satisfies GP.

GGG(n,p) - G is a random graph with G(n,p) - G is a random graph with n vertices and edge probability p.n vertices and edge probability p.

Main TheoremMain Theorem

Let GNP be any monotone property Let GNP be any monotone property of graphs on n vertices . of graphs on n vertices .

If If pp(GNP) > (GNP) > then then

qq(GNP) > 1-(GNP) > 1- for q = p + for q = p + cc11log(1/2log(1/2)/log)/lognn

absolute constant

Example-Max CliqueExample-Max Clique

Consider GConsider GG(n,p).G(n,p). The length of the interval of The length of the interval of

probabilities pprobabilities p for which the clique for which the clique number of Gnumber of G is almost surely is almost surely k k (where (where k k log log nn) is of order log) is of order log-1-1n.n.

The threshold interval: The transition The threshold interval: The transition between clique numbers k-1 and k.between clique numbers k-1 and k.

Probability for choosing an edge

Number of vertices

The probability of having a (The probability of having a (k k + 1)-clique + 1)-clique is still small (is still small ( log log-1-1nn). ).

The value of pThe value of p must increase bymust increase by clogclog-1-1n n before the probability for having a (before the probability for having a (k k + 1)-+ 1)-clique reaches clique reaches and another transition and another transition interval begins.interval begins.

The probability of having The probability of having a clique of size ka clique of size k is is 1-1-

The probability of having The probability of having a clique of size ka clique of size k is is

Def: Sharp thresholdDef: 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

Conjecture

Let GNP be any monotone property Let GNP be any monotone property of graphs on n vertices. If of graphs on n vertices. If pp(GNP) > (GNP) > then then qq(GNP) > 1-(GNP) > 1- for q = p + for q = p + clog(1/2clog(1/2)/log)/log22nn


Every Monotone Graph Every Monotone Graph Property has a sharp Property has a sharp

thresholdthreshold

A graph property is a property of A graph property is a property of graphs which is closed under graphs which is closed under isomorphism.isomorphism.

hereditary :hereditary : Let P be a monotone graph property; that Let P be a monotone graph property; that

is, if a graph G satisfies Pis, if a graph G satisfies P Every graph H on the same set of vertices, Every graph H on the same set of vertices,

which contains G as a sub graph satisfies P which contains G as a sub graph satisfies P as well.as well.


Hereditary in Hereditary in 3-colorable graphs3-colorable graphs

Examples of graph Examples of graph propertiesproperties

G is connectedG is connected G is HamiltonianG is Hamiltonian G contains a clique of size tG contains a clique of size t G is not planarG is not planar The clique number of G is larger than The clique number of G is larger than

that of its complementthat of its complement the diameter of G is at most sthe diameter of G is at most s G admits a transitive orientationG admits a transitive orientation ... etc .... etc .


ModelModel Erdös - Rényi Erdös - Rényi for for random graphrandom graph Choose every edge with probability pChoose every edge with probability p


ModelModel Erdös - Rényi Erdös - Rényi for for random graphrandom graph

Choose every edge Choose every edge with probability pwith probability p

DefinitionsDefinitions

GNPGNP – a graph property – a graph property

((PP)) - the probability that a random - the probability that a random graph on n vertices with edge graph on n vertices with edge probability p satisfies GP. probability p satisfies GP.

GGG(n,p) - G is a random graph with G(n,p) - G is a random graph with n vertices and edge probability p.n vertices and edge probability p.

Example – max cliqueExample – max clique

Let GLet GG(n,p) G(n,p)

Sharp thresholdSharp 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

Mechanism DesignMechanism Design

Shortest Path ProblemShortest Path Problem

Mechanism Design Mechanism Design ProblemProblem N agentsN agents ,bidders,bidders, each agent i has, each agent i has privateprivate

input tinput tiiT. Everything else in this scenario isT. Everything else in this scenario is publicpublic knowledge.knowledge.

TheThe output specificationoutput specification maps to each type maps to each type vector t= tvector t= t1 1 …t…tnn a set of allowed outputs o a set of allowed outputs oO.O.

Each agent i has aEach agent i has a valuationvaluation for his items: for his items: VVii(t(tii,o) = outcome for the agents.,o) = outcome for the agents.Each agent wishes to optimize his own utility.Each agent wishes to optimize his own utility.

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

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

Truth implementationTruth implementation The action of an agent consists of reporting The action of an agent consists of reporting

its type, its true type.its type, its true type.

Playing the truth is the dominating strategyPlaying the truth is the dominating strategy

THMTHM: If there exists a mechanism then there : If there exists a mechanism then there exists also a Truthful Implementation.exists also a Truthful Implementation.

ProofProof: simulate the hypothetical : simulate the hypothetical implementationimplementationbased on the actions derived from the based on the actions derived from the reported types.reported types.

Vickery-Groves-Clarke Vickery-Groves-Clarke (VGC)(VGC)

Mechanism Design for SPMechanism Design for SP

50$

10$

50$

10$Always in the shortest

path

Shortest Path using VGCShortest Path using VGC

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

graph G and two vertices source s and target t.graph G and two vertices source s and target t. Agents Agents = edges in G= edges in G Each agent has a cost for sending a single Each agent has a cost for sending a single

message on his edge denote by tmessage on his edge denote by tee..

ObjectiveObjective:: find the shortest (cheapest) path find the shortest (cheapest) path from s to t.from s to t.

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

C(G)C(G) = costs along the shortest path = costs along the shortest path (s,t) in G.(s,t) in G.

compute a shortest path in the G , at compute a shortest path in the G , at cost C(G) . cost C(G) .

Each agent that participates in the SP Each agent that participates in the SP obtains the payment she demanded obtains the payment she demanded plus plus [ C(G\e) – t[ C(G\e) – tee ]. ].

Shortest Path using VGCShortest Path using VGC

SP on G\e

How much will we pay?How much will we pay?

50$

10$

50$

10$

juntajunta

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

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

k

SS k

ff S

k

SS k

ff S

k

SS k

ff S

k

SS k

ff S

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 J J has little influence has little influence

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

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

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

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 of Show the complement of JJ has little influence has little influence

O as f / εj = 2 O as f / εj = 2

AltogetherAltogether

k kJ

i J

2 2

O(k)S S

i S,S k i Si J i J r2

4/ r

O(k)S

i Si J 2

22/ rO(k) O(k) r

i J

ff

f(S) 2 f(S)

2 f(S)

as f2 f 2

i

i

influence influence

influence

k kJ

i J

2 2

O(k)S S

i S,S k i Si J i J r2

4/ r

O(k)S

i Si J 2

22/ rO(k) O(k) r

i J

ff

f(S) 2 f(S)

2 f(S)

as f2 f 2

i

i

influence influence

influence

Beckner

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