Analysis of Boolean FunctionsAnalysis of Boolean FunctionsandandComplexity TheoryComplexity TheoryEconomicsEconomicsCombinatoricsCombinatoricsEtc.Etc.

Analysis of Boolean Functions Analysis of Boolean Functions

and and

Complexity Theory Complexity Theory

Economics Economics

Combinatorics Combinatorics

Etc. Etc.

Slides prepared with help of Ricky Rosen

Slides prepared with help of Ricky Rosen

Influential

Influential People People



The theory of the The theory of the Influence Influence of Variables on of Variables on Boolean Functions

Boolean Functions [ [ KKL KKL ,BL,R,M] ,BL,R,M] , has been , has been introduced to tackle

introduced to tackle Social Choice Social Choice problems problems and and distributed computing distributed computing . .



It has motivated a magnificent body of work, It has motivated a magnificent body of work, related to

related to



Sharp Threshold Sharp Threshold [F, FK] [F, FK]



Percolation Percolation [BKS] [BKS]



Economics: Economics: Arrow’s Theorem Arrow’s Theorem [K] [K]



Hardness of Approximation Hardness of Approximation [DS] [DS]

Utilizing

Utilizing Harmonic Analysis of Boolean functions Harmonic Analysis of Boolean functions … …



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

Where to go for Dinner?

Where to go for Dinner?

The The alternatives alternatives

Diners would cast their vote Diners would cast their vote

in an (electronic) envelope in an (electronic) envelope The system would decide – The system would decide –

not necessarily according to not necessarily according to

majority…

majority…

And what if And what if

someone someone

(in Florida?) (in Florida?)

can flip can flip

some votes some votes

Power Power

influence

influence

  ^0,1

f : P[n] ^    ^0,1

f : P[n]

Boolean Functions Boolean Functions

 Def Def : : A A Boolean function Boolean function

 

 

[ ] [ ]

 1,1









P n  x n 

 

[ ] [ ]

 1,1









P n x n

Power 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 Sensitivity Noise 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 out It turns out : one cannot design an : one cannot design an f f that would be robust to such that would be robust to such

noise --that is, would, on average, noise --that is, would, on average,

change value w.p.

change value w.p. < <   ^{O(1) O(1)} -- unless -- unless

determining the outcome according determining the outcome according

to very few of the voters

to very few of the voters

1-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

-1

 Def Def : : the the influence influence of of i i on on f f is the is the probability, over a random input

probability, over a random input x x , that , that f f changes its value when

changes its value when i i is flipped is flipped

Voting and

Voting and influence influence

  ^

_{   } ^ _  ^{  }  ^  ^{ }  ^ _

i

f

Pr f x i f x \ i

influence

  f ^

Pr f x _{   } ^ _  ^{  } i  ^ f x \ i  ^{ }  ^ _

influence



The The influence influence of of i i on on Majority Majority is the probability, is the probability, over a random input

over a random input x x , , Majority Majority changes with changes with i i



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

(people) vote -1 -1 and half vote and half vote 1 1 . .



i.e. i.e.

 Majority Majority :{1,-1} :{1,-1}

  { { 1 1 , , -1 -1 } }

 

1 1

2 1 / 2

influence i ^   

      

n n

n n

 

1 1

2 1 / 2

influence i ^   

      

n n

n n

1 -1 -1 ? 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1

-1

 Parity Parity : : {1,-1} {1,-1}

  { { 1 1 , , -1 -1 } }

 

 



 ^{n i i}  ^{n j}

i 1 j i

i

Parity(X) x x x

1 Influence

 

 



 ^{n i i}  ^{n j}

i 1 j i

i

Parity(X) x x x

1 Influence

Always changes the

value of

parity

 influence of influence of i i on on Dictatorship Dictatorship

= 1 = 1 . .

 influence of influence of j j   i i on on Dictatorship Dictatorship

= = 0 0 . .

 Dictatorship Dictatorship

:{1,-1} :{1,-1}

  { { 1 1 , , -1 -1 } }

 Dictatorship Dictatorship

(x)=x (x)=x

1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

-1

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

 Def Def : : the the Average Sensitivity Average Sensitivity of of f f ( ( as as ) ) is the sum of influences of all

is the sum of influences of all coordinates

coordinates i i   [n] [n] : :

 as as (Majority) = O(n (Majority) = O(n ^{½ ½} ) )

 as as (Parity) = n (Parity) = n

 as as (dictatorship) =1 (dictatorship) =1

  ^  ⁱ  

i

f f

as   ^  influence ⁱ  

i

f f

as influence

When

When as as (f)=1 (f)=1

Def Def : : f f is a is a balanced balanced function if it equals function if it equals -1 -1 exactly half of the times:

exactly half of the times:

E E

[f(x)]=0 [f(x)]=0 Can a balanced

Can a balanced f f have have as as (f) < 1 (f) < 1 ? _? What about

What about as as (f)=1 (f)=1 ? _? Beside dictatorships?

Beside dictatorships?

Prop Prop : : f f is is balanced balanced and and as as (f)=1 (f)=1

  f f is a is a dictatorship dictatorship . .

Representing

Representing f f as a Polynomial as a Polynomial

 What would be the monomials over What would be the monomials over x x   P[n] P[n] ? ?

 All powers except All powers except 0 0 and and 1 1 cancel out! cancel out!

 Hence, one for each Hence, one for each character character S S   [n] [n]

 These are all the These are all the multiplicative functions multiplicative functions

 

 ^





 

 ^{S x}

S i

i S

(x) x   1

 ^





 

 ^{S x}

S i

i S

(x) x 1

Fourier-Walsh Transform Fourier-Walsh Transform

 Consider all characters Consider all characters

 Given any function Given any function

let the Fourier-Walsh coefficients of

let the Fourier-Walsh coefficients of f f be be

 thus thus f f can be described as can be described as

  ^{ }

f : P n   ^{ }

f : P n





 

S i

i S

(x) x





 

S i

i S

(x) x

  _S   _S  

f S     f   _S  E f x x       _S   x   f S   f    E f x x     x  

  ^

  ^{ S}

S

f ^  f S ^   ^{ S}

S

f f S

Norms Norms

Def Def : : Expectation Expectation norm on the function norm on the function Def Def : : Summation Summation norm on the transform norm on the transform

Thm Thm [Parseval]: [Parseval]:

Hence

Hence , for a Boolean , for a Boolean f f

 

 

   

q q

q x P[n]

f f(x)

 

 

   

q q

q x P[n]

f f(x)

 

 

 ^q   ^q

q S n

f f S  

 

 ^q   ^q

q S n

f f S

 

2 2

f   f

2 2

f f

 

 ^{ 2 2} ₂

S

f (S)  f  1

 ^{ 2 2} ₂

S

f (S) f 1

Simple

Simple Observations Observations

 Def Def : :

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

{-1,0,1} : :

  

   

1 x P[n]

f f(x)

  

   

1 x P[n]

f f(x)

        

q 1

q 1 x P[n]

f f Pr f(x) { 1,1}

        

q 1

q 1 x P[n]

f f Pr f(x) { 1,1}

Variables` Influence Variables` Influence

 Recall: Recall: influence influence of an index of an index i i   [n] [n] on a on a Boolean function

Boolean function f:{1,-1} f:{1,-1}

  {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 f f Claim

Claim : :

 And the as: And the as:

     

  



 

 

   

 _{x P n}

(f) Pr f x f x i Influence _i i ^{(f) } _{x P n } Pr f x f x i

^ _{ }   ^  ^    ^ _{ }

Influence

  ^{ }



  ^{ 2}

S,i S

f f S

Influence i   ^{ }



  ^{ 2}

S,i S

f f S

Influence i

 

     ^{ 2}

S

f = f S S

as ^{ } _{ }  ^{ 2}  

S

f = f S S

as

Fourier Representation of Fourier Representation of

influence influence

Proof

Proof : consider the influence : consider the influence function function

which in Fourier representation is which in Fourier representation is

and and

     

i

f x f x i

f x 2

 

      

i

f x f x i

f x 2

 



       

 

i S S S

S S

i S S

1 1

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

2 2

f(S) x

  





    



  



       

 

i S S S

S S

i S S

1 1

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

2 2

f(S) x

  





    



  



  ^{  2 2}

i i 2

i S

f f x f (S)



   ^

influence i   i ^{  2} 2 ² i S

f f x f (S)



   ^

influence

Balanced

Balanced f f s.t. s.t. as as (f)=1 (f)=1 is Dict. is Dict.

 Since Since f f is balanced and is balanced and

 So So f f is linear is linear

 For any For any i i s.t. s.t.

  ^{ }

f   ^{ } 0

f 0

     





  





S S

ˆ ˆ

f S S   f S S   as   f 1





  





S S

ˆ ˆ

f S S f S S as f 1

    _{ }

 ^

i

f =  f iχ ^     _{ }

i

f = f iχ

If s s.t |s|>1 and then as(f)>1

  

f s   0

f s 0

  ^

f {i}   ^ 0

f {i} 0

      ^{ }

  ^{ }

    

  

   





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

f { i } 1 ,1 f x or f x

      ^{ }

  ^{ }

    

  

   





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

f { i } 1 ,1 f x or f x

Only i has

changed

Expectation and Variance Expectation and Variance

 Claim Claim : :

 Hence, for any Hence, for any f f

  x

f       E f(x)    f    E f(x) x    

   

     

   

2 2

x P n x P n

2 2

2

2 S n,S

f f x E f x

f f f S

V _{ } E

 

      

 



 

 ^   ^

   

     

   

2 2

x P n x P n

2 2

2

2 S n,S

f f x E f x

f f 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 Percolation First Passage Percolation

 Consider the Grid Consider the Grid

 For each edge For each edge e e of choose of choose independently independently w w

= 1 = 1 or or w w

= 2 = 2 , each with probability , each with probability ½ ½

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

 Thm Thm : The variance of the shortest path : The variance of the shortest path from the origin to vertex

from the origin to vertex v v is bounded from is bounded from above by

above by O( |v|/ log |v|) O( |v|/ log |v|) [BKS] [BKS]

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

Z

Z Z

Z

Z

Z

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

invariant under isomorphism.

Def Def : : a a monotone monotone graph property is a graph graph property is a graph property

property P P s.t. s.t.

 If If P(G) P(G) then for every super-graph then for every super-graph H H of G of G

(namely, a graph on the same set of vertices, (namely, a graph on the same set of vertices,

which contains all edges of

which contains all edges of G G ) ) P(H) P(H) as well. as well.

P P is in fact a Boolean function: is in fact a Boolean function:

P: {-1, 1}

P: {-1, 1} ^{V V}

  {-1, 1} {-1, 1}

Graph properties

Graph properties

Examples of graph properties Examples of graph properties



G G is connected is connected



G G is Hamiltonian is Hamiltonian



G G contains a clique of size contains a clique of size t t



G G is not planar is not planar



The clique number of The clique number of G G is larger than that of is larger than that of its complement

its complement



The diameter of The diameter of G G is at most is at most s s



... etc . ... etc .

 What is the What is the influence influence of different of different e e on on

P P ? ?

Erdös–Rényi

Erdös–Rényi G(n,p) G(n,p) ^Graph _Graph

The The Erdös-Rényi Erdös-Rényi distribution of distribution of random graphs random graphs Put an edge between any two vertices w.p.

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

Definitions Definitions

 P P – a graph property – a graph property

   _{p p} (P) (P) - the probability that a - the probability that a random graph on

random graph on n n vertices with vertices with edge probability

edge probability p p satisfies satisfies P P . .

 G G   G(n,p) G(n,p) - - G G is a random graph of is a random graph of n n vertices and edge probability

vertices and edge probability p p . .

Def Def : Sharp threshold : Sharp threshold

 Sharp threshold in monotone graph Sharp threshold in monotone graph property:

property:

 The transition from a property being very The transition from a property being very unlikely to it being very likely is

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

G satisfies property P

G Does not

satisfies

property P

Thm Thm : : every monotone graph property every monotone graph property has a Sharp Threshold

has a Sharp Threshold [FK] [FK]

 Let Let P P be any monotone property of be any monotone property of graphs on

graphs on n n vertices . vertices . If If   _{p p (P) > (P) >}   then then

  _{q q} (P) > 1- (P) > 1-   for for q q = = p + c p + c 1 ₁ log(½ log(½   )/log )/log n n Proof idea

Proof idea : show : show as as p’ _p’ (P) (P) , for , for p’>p p’>p , is high , is high

Thm Thm [Margulis-Russo]: [Margulis-Russo]:

For monotone For monotone f f

q q

d (f) (f) dq

 

as _q d (f) ^q (f) dq

 

as

Proof

Proof [Margulis-Russo]: [Margulis-Russo]:

  ⁿ

  ⁿ  

q q q

i q

i 1 i i 1

d f f

f (f)

dq  q 

  

 

   ^{influence as}

  ⁿ

  ⁿ  

q q q

i q

i 1 i i 1

d f f

f (f)

dq  q 

  

 

   ^{influence as}

Mechanism Design Problem Mechanism Design Problem



N N agents agents , each agent , each agent i i has has private private input input t t

  T T . All . All other information is

other information is public public knowledge. knowledge.



Each agent Each agent i i has a has a valuation valuation for all items: for all items:

Each agent wishes to optimize her own utility.

Each agent wishes to optimize her own utility.



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

total payment.



Means Means : protocol between agents and auctioneer : protocol between agents and auctioneer . .

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

 Sealed bid auction Sealed bid auction

 A A Truth Revealing Truth Revealing protocol, namely, one protocol, namely, one in which each agent might as well reveal in which each agent might as well reveal

her valuation to the auctioneer her valuation to the auctioneer

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

win the auction

win the auction

Shortest Path using VGC Shortest Path using VGC

 Problem definition: Problem definition:



Communication network Communication network modeled by a directed modeled by a directed graph

graph G G and two vertices source and two vertices source s s and target and target t t . .



Agents Agents = edges in = edges in G G



Each agent has a cost for sending a single message Each agent has a cost for sending a single message on her edge denote by

on her edge denote by t t

. .



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



Means Means : : protocol between agents and auctioneer. protocol between agents and auctioneer.

VCG for Shortest-Path VCG for Shortest-Path

50$

10$

50$

10$ Always in the shortest

path

How much will we overpay?

How much will we overpay?

 SP SP

 Every agent gets an extra $1 Every agent gets an extra $1

 Thm Thm [Mahedia,Saberi,S]: expected extra [Mahedia,Saberi,S]: expected extra pay is

pay is as as

(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 Functions Learning Functions

 To To learn learn a function a function f:G f:G

  G G is to sample it in is to sample it in poly(n|G|/

poly(n|G|/   ) ) points and come up with some points and come up with some f’ f’

that differs with

that differs with f f on at most an on at most an   fraction of fraction of the points

the points



Membership Membership : choose your points : choose your points



Random Random : w.h.p. over values for a small : w.h.p. over values for a small random random set set of points

of points

 Applications (when learnable): bioinformatics, Applications (when learnable): bioinformatics, economy, etc.

economy, etc.

 Also [Akavia, Goldwasser, S.]: cryptography -- Also [Akavia, Goldwasser, S.]: cryptography -- hardcore predicates via list-decoding

hardcore predicates via list-decoding

Concentrated Concentrated

 Def Def : the : the restriction restriction of of f f to to   is is

 Def Def : : f f is a is a concentrated function concentrated 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} ^{n n}   {0,1} {0,1} concentrated is learnable concentrated is learnable

 Thm[Akavia, Goldwasser, S.]: over any Thm[Akavia, Goldwasser, S.]: over any Abelian group

Abelian group f:G f:G ^{n n}   G G







  ^

| S:S

f

f(S) 



  ^

| S:S

f f(S)







 f f 

  f f

characters

…-5 -3 -1 1 3 5…

Juntas Juntas

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

depends on only J J variables. variables.



A Dictatorship is 1-junta A Dictatorship is 1-junta

1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 1

1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

-1 -1

Juntas Juntas

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

depends on only J J variables. variables.

 Thm Thm [Fischer, Kindler, Ron, Samo., S]: [Fischer, Kindler, Ron, Samo., S]:

Juntas are

Juntas are testable testable

 Thm Thm [ [ Kushilevitz, Mansour; Mossel, Odonel Kushilevitz, Mansour; Mossel, Odonel ]: ]:

Juntas are

Juntas are learnable learnable

1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1 -1 1 1 -1 1 -1 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  

distribution. distribution.

Choose a subset, I, of variables Each var is in the set with probability Choose a subset, I, of variables Each var is in the set with probability 

1-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1

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?

What is the new value of f?

  ^ _ ^{  }    ^  ^ _

       

NS ,p f  x n p ,I Pr n ,z I p f x f x \ I z

  ^ _ ^{  }  ^{  }  ^ _

       

NS ,p f  x n p ,I Pr n ,z I p f x f x \ I z

Noise sensitivity and juntas Noise sensitivity and juntas



Juntas are noise insensitive (stable) Juntas are noise insensitive (stable) Thm Thm [Bourgain; Kindler & S]: [Bourgain; Kindler & S]:

Noise insensitive (stable) Boolean functions are Juntas Noise insensitive (stable) Boolean functions are Juntas

1-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1

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

Junta

Junta

Freidgut Theorem Freidgut Theorem

Thm Thm : any Boolean : any Boolean f f is an is an [ [   , j]- , j]- junta for junta for

Proof Proof : :

1.1.

Specify the junta Specify the junta J J

2.2.

Show the complement of Show the complement of J J has little influence has little influence

   ^{f /} 

j = 2 ^{O as} O as ^{   f /  }

j = 2 ^

Long-Code Long-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 dictatorships monotone 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 n n elements elements

Long-Code Long-Code

 Encoding an element Encoding an element e e   [n] [n] : :

 E E

legally-encodes legally-encodes an element an element e e if if E E

= f = f

F F F F T T T T T T

Open Questions Open Questions



Hardness of Approximation Hardness of Approximation : :



MAX-CUT MAX-CUT



Coloring Coloring a 3-colorable graph with fewest colors a 3-colorable graph with fewest colors



Graph Properties Graph Properties : find sharp-thresholds for : find sharp-thresholds for properties

properties



Circuit Complexity Circuit Complexity : switching lemmas : switching lemmas



Mechanism Design Mechanism Design : show a non truth-revealing : show a non truth-revealing

protocol in which the pay is smaller (Nash equilibrium protocol in which the pay is smaller (Nash equilibrium

when all agents tell the truth?) when all agents tell the truth?)



Analysis Analysis : show weakest condition for a function to be : show weakest condition for a function to be a Junta

a Junta



Learning Learning : by random queries : by random queries



Apply Apply Concentration of Measure Concentration of Measure techniques to other techniques to other problems in Complexity Theory

problems in Complexity Theory

Where to go for Dinner?

Where to go for Dinner?

The The alternatives alternatives

Diners would cast their vote Diners would cast their vote

in an (electronic) envelope in an (electronic) envelope The system would decide – The system would decide –

not necessarily according to not necessarily according to

majority…

majority…

And what if And what if

someone someone

(in Florida?) (in Florida?)

can flip can flip

some votes some votes

Power Power

influence influence

Of course they’ll have to discuss it over

dinner….

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

Corollary Corollary : :

fix a

fix a p p -biased distribution -biased distribution   _{p p} over over P([n]) P([n]) Let Let   >0 >0 be any parameter. be any parameter.

Set Set k=log k=log 1- _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 -2} k k ^{3 3}   ^{2k 2k} ) )

 

k 2 2 2

f

 O k   

f

 O k 

   

ns f

   O k   

ns f

 O k 

 Def Def ( (    _ ,p,x ,p,x [n] [n] ): Let ): Let 0< 0<   <1 <1 , and , and x x   P([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~  

is a is a noise subset noise subset , and , and



z~  z~ 

is a is a replacement replacement . .

Def Def ( (   - - noise-sensitivity noise-sensitivity ): let ): let 0< 0<   <1 <1 , then , then

[ When

[ When p=½ p=½ equivalent to flipping each equivalent to flipping each coordinate in

coordinate in x x independently w.p. independently w.p.   /2 /2 .] .]

   

[n] [n]

p ,p,x

x~ ,y~

ns f = Pr f x f y





   

        

p ,p,x

x~ ,y~

ns f = Pr f x f y





   

    

Noise-Sensitivity

Noise-Sensitivity

Noise-Sensitivity – Cont.

Noise-Sensitivity – Cont.

 Advantage Advantage : very efficiently testable (using : very efficiently testable (using only two queries) by a

only two queries) by a perturbation-test perturbation-test . .

 Def Def ( ( perturbation-test perturbation-test ): choose ): choose x~ x~  

, and , and y~ y~  

, 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 f f . .

 Prop Prop : the : the   -noise-sensitivity is given by -noise-sensitivity is given by

 

 

S

2 ns f =1 

      ^ 1   ^

f S ^

^{ }

S

2 ns f =1 

      1   f S ^

Relation between Parameters Relation between Parameters

Prop Prop : small : small ns ns ^{ } small _small high-freq weight high-freq weight Proof

Proof : :

therefore:

therefore:

if if ns ns is small, then is small, then Hence the

Hence the high frequencies high frequencies must have must have small weights (as

small weights (as ). ).

Prop Prop : small : small as as   small small high-freq weight high-freq weight Proof

Proof : :  

S

f   f (S) S ^ as  

S

f   f (S) S ^ as

 

 

S

2 ns f =1 

      ^ 1   ^

f S ^

^{ }

S

2 ns f =1 

      1   f S ^

 

 

S

1   f S ~ 1

 ^{ }

^

^{ }

S

1   f S ~ 1

 ^

2

 

S

f S 1 

 ^

^{ }

S

f S 1 

 ^

High vs. Low Frequencies High vs. Low Frequencies

Def Def : The section of a function : The section of a function f f above above k k is is

and the

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

 

k S

S k

f ^ f S 

  ^

  

k S

S k

f ^ f S 

  ^



 

k S

S k

f ^ f S 

  ^

  

k S

S k

f ^ f S 

  ^



Low-degree B.f are Juntas Low-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(  

k k

 

) ) Corollary

Corollary : : fix a

fix a p p -biased distribution -biased distribution  

over over P([n]) P([n]) Let Let   >0 >0 be any parameter. be any parameter.

Set Set k=log k=log

(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(  

k k

 

) )

 

k 2 2 2

f

 O k   

f

 O k 

   

ns f

   O k   

ns f

 O k 

Specify the Junta Specify 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 let and let hence,

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

   ^ 

 _i 

J ^  i| influence ⁱ   f ^{ } 

J i| influence f

  

    ²

J 2

A f _{   } ² _{ } 1  2

J 2

A f 1 2

 

 

 J     

f'(x) sign A f x J ^  J ^{ }    ^  

f'(x) sign A f x J

Hadamard Code Hadamard Code

In the Hadamard code the In the Hadamard code the

set of legal-words consists of set of legal-words consists of

all multiplicative (linear if over all multiplicative (linear if over

{0,1}

{0,1} ) functions ) functions

C={ C={   _{S S} | S | S   [n]} [n]}

namely all characters

namely all characters

76

Hadamard Test – Soundness Hadamard Test – Soundness

Prop Prop (soundness): (soundness):

Proof Proof : :

     

     

     

 

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 S

S ,S ,S

1 2 3 x,y S S S S

S ,S ,S

1 2 3 x S S y S S

S ,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-code Testing Long-code

Def Def (a (a long-code list-test long-code list-test ): given a code-word ): given a code-word f f , , probe it in a constant number of entries, and probe it in a constant number of entries, and



accept almost always if accept almost always if f f is a monotone is a monotone dictatorship

dictatorship



reject w.h.p if reject w.h.p if f f does not have does not have a sizeable fraction a sizeable fraction of its Fourier weight concentrated on a small set of of its Fourier weight concentrated on a small set of

variables, that is, if

variables, that is, if   a a semi-Junta semi-Junta J J   [n] [n] s.t. s.t.

Note Note : a long-code list-test, distinguishes : a long-code list-test, distinguishes between the case

between the case f f is a is a dictatorship dictatorship , to the , to the case

case f f is far from a is far from a junta junta . .

2

 

S J

f S 



 ^

^{ } 

S J

f S 



 ^ 

Motivation – Testing Long-code Motivation – Testing Long-code

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

proving hardness results.

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

a function to be a junta is important.

High Frequencies Contribute Little High Frequencies Contribute Little

Prop Prop : : k >> r log r k >> r log r implies implies

Proof

Proof : a character : a character S S of size larger than of size larger than k k spreads w.h.p. over all parts

spreads w.h.p. over all parts I I

, 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 ), then surely there would be more than

surely there would be more than j j parts parts I I

that fail the

that fail the t t independence-tests independence-tests

  ^





  ^{ 2} 

k 2

2 S k

f ^ f S   ^ 4



  ^{ 2} 

k 2

2 S k

f f S 4

Altogether Altogether

Lemma Lemma : :

Proof

Proof : : ^{influence influence}

    ^{f f    } _{2 2}

  ^{  k 2  k}   ^{ }

J f f 2 J f 2

influence _J   ^{f  f  k 2} ₂ +influence ^ _J ^k   ^{f  } ₂

influence +influence

Altogether Altogether

   



 



 





 



  ^

k k

J i J

2

i S, S k S

i J 2

f f

f(S) ?

influence   influence i  



 



 





 



  ^

k k

J i J

2

i S, S k S

i J 2

f f

f(S) ?

influence influence i

Beckner/Nelson/Bonami Beckner/Nelson/Bonami

Inequality Inequality

Def Def : let : let T T

be the following operator on any be the following operator on any f f , ,

Prop Prop : : Proof Proof : :

 

 

 



  

   

     

1 /2

z

f x E f x z

T  

 

 



  

   

     

1 /2

z

f x E f x z

T

   

  



  

   ^{S  S}

S n

f f S

T

   

  



  

   ^{S  S}

S n

f f S

T

     

   

  



      

   ^{ S}  ^S

S n z

f x f S x E z

T      

   

  



      

   ^{ S}  ^S

S n z

f x f S x E z

T

Inequality Inequality

Def Def : let : let T T

be the following operator on any be the following operator on any f f , ,

Thm Thm : for any : for any p≥r p≥r and and   ≤((r-1)/(p-1)) ≤((r-1)/(p-1))

 

 

 



  

   

     

1 /2

z

f x E f x z

T  

 

 



  

   

     

1 /2

z

f x E f x z

T

     f _p  f _r

T _     f _p  f _r

T

Beckner/Nelson/Bonami Corollary Beckner/Nelson/Bonami Corollary

Corollary 1

Corollary 1 : for any real : for any real f f and and 2≥r≥1 2≥r≥1

Corollary 2

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

  ^

   ^{k 2} _r

2 r 1 f

f ^{ k} k ₂ ^  ^{r 1 }  ^{ k 2 f} _r

f

 

   ^{k 2} ₂

r r 1 f

f ^{ k} k _r ^  ^{r 1 }  ^{k 2 f} ₂

f

Perturbation Perturbation

Def Def : denote by : denote by   _{ } the the

distribution over all subsets of distribution over all subsets of [n] [n] , which assigns probability to , which assigns probability to

a subset

a subset x x as follows: as follows:

independently, for each

independently, for each i i   [n] [n] , let , let

 i i   x x with probability with probability 1- 1-  

 i i   x x with probability with probability  

Long-Code Test Long-Code Test

Given a Boolean

Given a Boolean f f , choose , choose random

random x x and and y y , and choose , and choose z z   _{ } ; check that ; check that

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

Prop Prop (completeness): a legal long- (completeness): a legal long- code word (a dictatorship)

code word (a dictatorship) passes this test w.p.

passes this test w.p. 1- 1-  

Long-code Tests Long-code Tests

 Def Def (a (a long-code test long-code test ): given a code- ): given a code- word

word w w , probe it in a constant number of , probe it in a constant number of entries, and

entries, 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 w w is not close to any is not close to any monotone dictatorship

monotone dictatorship

Efficient Long-code Tests Efficient Long-code Tests

For some applications, it suffices if the test may For some applications, it suffices if the test may

accept illegal code-words, nevertheless, ones accept illegal code-words, nevertheless, ones

which have short list-decoding:

which have short list-decoding:

Def Def (a (a long-code list-test long-code list-test ): given a code-word ): given a code-word w w , , 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 w w is not even is not even approximately approximately

determined by a short list of domain elements

determined by a short list of domain elements , that , that is, if

is, if   a a Junta Junta J J  [n] [n] s.t. s.t. f f is close to is close to f’ f’ and and f’(x)=f’(x

f’(x)=f’(x  J) J) for all for all x x

Note Note : a long-code list-test, distinguishes between : a long-code list-test, distinguishes between the case

the case w w is a is a dictatorship dictatorship , to the case , to the case w w is is far from a

far from a junta junta . .

Codes and Boolean Functions Codes and Boolean Functions

Def Def : an : an m m -bit code is a subset of the set of all -bit code is a subset of the set of all the the m m -binary string -binary string

C C   {-1,1} {-1,1}

The The distance distance of a code of a code C C is the minimum, over is the minimum, over all pairs of legal-words (in

all pairs of legal-words (in C C ), of the ), of the

Hamming distance between the two words Hamming distance between the two words Note Note : A Boolean function over : A Boolean function over n n binary binary

variables is a

variables is a 2 2

-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 2 2

-bits code -bits code

Long-Code

Long-Code   Monotone-Dictatorship Monotone-Dictatorship

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

are all monotone dictatorships

C={ C={   _{{i} {i}} | i | i   [n]} [n]}

namely, all the singleton characters

namely, all the singleton characters