Analysis of Boolean Functions Fourier Analysis, Projections, Influence, Junta, Etc… And (some) applications Slides prepared with help of Ricky Rosen.

Analysis of Boolean Analysis of Boolean FunctionsFunctions

Fourier Analysis, Fourier Analysis,Projections,Projections, Influence, Influence,

Junta,Junta,Etc… Etc…

And (some) applicationsAnd (some) applications Slides prepared with help of Ricky RosenSlides prepared with help of Ricky Rosen

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

Functions as Vector-Functions as Vector-SpacesSpaces

ff**

-1*-1*

1*1*

11*11*

11-1*

11-1*

-1-1*-1-1*

-11*-11*

-11-1*-11-1*

-111*

-111*

-1-1-1*-1-1-1*

-1-11*-1-11*

111*111*

1-1*1-1*1-1-1*

1-1-1*

1-11*

1-11*

ff2n2n*

*

-1*-1*

1*1*

11*11*

11-1*11-1*

-1-1*-1-1*

-11*-11*

-11-1*-11-1*

-111*-111*

-1-1-1*-1-1-1*

-1-11*-1-11*

111*111*

1-1*1-1*

1-1-1*1-1-1*

1-11*1-11*

A function can be represented as a A function can be represented as a string of size string of size 2n (i.e.: it’s truth table)(i.e.: it’s truth table)

Functions’ Vector-Space Functions’ Vector-Space

A functions A functions ff is a vector is a vector

Addition:Addition: ‘f+g’(x) = f(x) + g(x)‘f+g’(x) = f(x) + g(x)

Multiplication by scalarMultiplication by scalar

‘‘ccf’(x) = cf’(x) = cf(x)f(x)

Inner product (normalized)Inner product (normalized)

n2f n2f

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*

nx 2

f g f x g xE

nx 2

f g f x g xE

Boolean function as voting Boolean function as voting system system

Consider Consider nn agents, each voting either agents, each voting either “for” (“for” (T=-1T=-1) or “against” () or “against” (F=1F=1) )

The system is not necessarily The system is not necessarily majoritymajority..

This is a This is a boolean functionboolean function over over nn variablesvariables..

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 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

X represented as a set of variablesX represented as a set of variables

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

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

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 Sx

f S f E f x x

S

S

ff S S

S

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

1x 1x

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

2x 2x

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

Distribution over CharactersDistribution over Characters

Characters and Characters and MultiplicativeMultiplicative

ClaimClaim:: Characters are all the Characters are all the multiplicative functionsmultiplicative functions

ProofProof: :

Let Let S={i | f({i})=-1 }S={i | f({i})=-1 } we provewe prove (f = (f = ss))

2ff f × f ×= f 1= = = 2ff f × f ×= f 1= = =

F is multiplicative function

F is multiplicative function

i 1 i

s x

ix i,f 1

= = =if x f x 1

i 1 i

s x

ix i,f 1

= = =if x f x 1

= =f ×ff 1 f -1,1 x x x x x = =f ×ff 1 f -1,1 x x x x x

SimpleSimple ObservationsObservations

DefDef::

For any function For any function ff whose range is whose range 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

Restriction and AverageRestriction and AverageDefDef: Let : Let II[n], x[n], xP([n]\I),P([n]\I),

the the restriction functionrestriction function isis

I

I

f x : P I 1,1

f x y f x y

I

I

f x : P I 1,1

f x y f x y

[n]I

x

y

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

II

xP[ [n]\I ]

Average functionAverage functionDefDef: the : the average functionaverage function isis

NoteNote::

I

Iy P I

A f : P I

A f x E f x y

I

Iy P I

A f : P I

A f x E f x y

I Iy P I

A f x E f x y

I Iy P I

A f x E f x y

[n]I

x

y y

y yy

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

II

xP[ [n]\I ]

In Fourier ExpansionIn Fourier Expansion

PropProp: :

FFII[x][x] is a functions only of the variables of is a functions only of the variables of II (since (since xxP[ [n]\IP[ [n]\I ] is fixed). ] is fixed).

Representing it as a polynomial hence Representing it as a polynomial hence involves coefficient only to involves coefficient only to S S I I , , , , each of which is the sum of all coefficient each of which is the sum of all coefficient of characters whose intersection with of characters whose intersection with II is is S S where the value is calculated according to where the value is calculated according to the restrictionthe restriction x x

I STS I T I S

f x f T x

I STS I T I S

f x f T x

f S f S

In Fourier ExpansionIn Fourier Expansion

RecallRecall: :

Since the expectation of a function is the Since the expectation of a function is the coefficient of its empty character:coefficient of its empty character:

Cor 1Cor 1::

Cor 2Cor 2::

I S

S I

A ff (S)

I S

S I

A ff (S)

I STS I T I S

f x f T x

I STS I T I S

f x f T x

2 2

i 2S,i S

f 1 A ff SiInfluence

2 2

i 2S,i S

f 1 A ff SiInfluence

P[{i}] = { ,{i} }

A{i}[x] {-1,0,1}

P[{i}] = { ,{i} }

A{i}[x] {-1,0,1}

Parseval + corollary 1 + the sum of squares of the coefficients of a boolean function

equals 1

Parseval + corollary 1 + the sum of squares of the coefficients of a boolean function

equals 1

Expectation and VarianceExpectation and Variance

RecallRecall::

Hence, for any Hence, for any ff

xf 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

Var E

22

x P n x P n

2 22

2S n,S

ff x E f x

ff f S

Var E

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 homogeneous & linear is homogeneous & 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

i

i

f = fi χ 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