Analysis of Boolean Functions
and
Complexity Theory
Economics
Combinatorics
…
Introduction

Objectives:


To introduce Analysis of Boolean
Functions and some of its
applications.
Overview:
Basic definitions.
 First passage percolation
 Mechanism design
 Graph property
 … And more…

Boolean Functions

Def: A Boolean function
f : P[n]  0,1
1,1

Power set
of [n]
Choose the
location of -1
Choose a sequence
of -1 and 1
P[n]   x  [n]
 1,1
n
1,4
 1,1,1, 1
Functions as Vector-Spaces
11*
11*
1*
1*
1-1*
1-1*
**
-11*
-11*
111*
111*
11-1*
11-1*
1-11*
1-11*
1-1-1*
1-1-1*
-111*
-111*
-11-1*
-11-1*
-1*
-1*
-1-1*
-1-11*
-1-11*
-1-1*
-1-1-1*
-1-1-1*
ff
2n
Functions’ Vector-Space

A functions f is a vector

Addition:
f 
‘f+g’(x) = f(x) + g(x)

2n
Multiplication by scalar
‘cf’(x) = cf(x)
Variable influence

f:{1,-1}20 {1,-1}
-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1
1
-1

influence of i on f is the probability that f
changes its value when i is flipped in a random
input x.
-1 1 -1 -1 ? 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1



Majority :{1,-1}19 {1,-1}
influence of i on Majority is the probability that
Majority changes its value when i is flipped in a
random input x this happens when half of the n-1
coordinate (people) vote -1 and half vote 1.
i.e.
 n 1 

2
 n  1 / 2 

1
influencei 

n
n
2
-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

Parity :{1,-1}20 {1,-1}
Parity(X ) 
n
x
i 1
Influencei  1
i
n
 xi  x j
j i
Always
changes the
value of
parity
-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1




Dictatorshipi :{1,-1}20 {1,-1}
Dictatorshipi(X)=xi
influence of i on Dictatorshipi= 1.
influence of ji on Dictatorshipi= 0.
Variables` Influence

The influence of a coordinate i [n] on a
Boolean function f:{1,-1}n {1,-1} is
Influencei
Pr f(x)  f(x  i ) 
The influence of i on f is the probability, over
a random input x, that f changes its value
when i is flipped.
Variables` Influence

Average Sensitivity of f (AS) - The sum
of influences of all coordinates i  [n].


# f ( x)  f ( x  i )
i

Average Sensitivity of f is the expected
number of coordinates, for a random input
x, flipping of which changes the value of f.
example



majority for
f :{1,1}  {1,1}
3
What is Average Sensitivity ?
AS= ½+ ½+ ½= 1.5
11*
11*
Monomials
1*
1*
10*
10*
**
01*
01*
0*
0*
00*
00*
111*
111*
110*
110*
101*
101*
100*
100*
011*
011*
010*
010*
001*
001*
000*
000*

What would be the monomials over x  P[n]?

All powers except 0 and 1 disappear!

Hence, one for each character S[n]
S (x)

 x   1
iS
Sx
i
These are all the multiplicative functions
ff
11*
11*
Fourier-Walsh Transform

Consider all characters
1*
1*
10*
10*
**
01*
01*
0*
0*
00*
00*
S (x)   xi
iS

Given any function f : P n 
let the Fourier-Walsh coefficients of f be
f  S   f  S

thus f can described as
f   f S S
S
111*
111*
110*
110*
101*
101*
100*
100*
011*
011*
010*
010*
001*
001*
000*
000*
ff
Norms

Def: Expectation norm on the function
f


q

q
  f (x ) 

x P [n ] 

1
q
Def: Summation Norm on its Fourier
transform
f

 f (x )  
  



 S [n ] 
q
q
1
q
11*
11*
Fourier Transform: Norm
Norm: (Sum)
f
Thm [Parseval]:
q
q
 f S 
1*
1*
10*
10*
**
01*
01*
0*
0*
00*
00*
q
Sn
f  f2
Hence, for a Boolean f
2
2
 f (S) 
S
2
f2 1
111*
111*
110*
110*
101*
101*
100*
100*
011*
011*
010*
010*
001*
001*
000*
000*
ff
2
 f (S)  1
S

We may think of the Transform as
defining a distribution over the
characters.

x
1
x
2
2
 f (S)  1
S
 x x ...x
1 2
n
Inner Product

Recall f 

Inner product (normalized)
2n
f ( x)  g ( x) 

xP[ n ]
f ,g  f g  
Simple Observations

Claim:

For any function f whose range is {-1,0,1}:
f
p
p
 f



f
(
x
)



1
xP[ n ]
f
1
1
f ( x) {1,1}

xP[ n ]
 Pr
Variables` Influence

Recall: influence of an index i [n] on a
Boolean function f:{1,-1}n {1,-1} is
Influencei (f)

Pr f  x   f  x 
i 
xPn 
Which can be expressed in terms of the
Fourier coefficients of f
Claim:
Influencei  f 
2
 f S
S,iS
Average Sensitivity




Def: the sensitivity of x w.r.t. f is # f  x   f  x 
i 
i
Thinking of the discrete n-dimensional
cube, color each vertex n in color 1 or color
-1 (color f(n)).
Edge whose vertices are colored with the
same color is called monotone.
The average sensitivity is the number of
edges whom are not monotone..
Average Sensitivity

Claim: as  f    fˆ 2  s  s
s

Proof:
as f = 
i
2
ˆ
 f S 
S|iS
2
ˆ
=  f S  S
S
When AS(f)=1





Def: f is a balanced function if E f(x)   0
x
THM: f is balanced
and as(f)=1  f is dictatorship.
because only x can change
the value of f
Proof:
 x, sens(x)=1, and as(f)=1 follows.
F is balanced since the dictator is 1
on half of the x and -1 on half of the
x.
When AS(f)=1
f is balanced


f   0
1 = as(f) =  fˆ2 (S) S =  fˆ2 (S) S
S


S 
So f is linear
For i whose
If s s.t |s|>1
and f  s   0
then as(f)>1
f =  fˆ i χi
i
f  {i}   0
f  x   f  x  i  2f  {i}   2,2
 f  {i}   1,1  f  xi or f  xi
Only i has
changed
First Passage Percolation




Consider the Grid
Zd
d
For each edge e of Z choose independently
we = a or we = b, each with probability ½ 0< a < b < .
This induces a random metric on the vertices of
Z
d
Proposition : The variance of the shortest path from
the origin to vertex v is bounded by O( |v| log |v|).
[BKS]
First Passage Percolation
Choose each edge with probability ½ to be 1
Proof sketch
First Passage Percolation




Consider the Grid
Zd
d
For each edge e of Z choose independently
we = 1 or we = 2, each with probability ½.
This induces a random metric on the vertices of
Z
d
Proposition : The variance of the shortest path from
the origin to vertex v is bounded by O( |v| log |v|).
Proof outline

Let G denote the grid

SPG – the shortest path in G from the origin to v.
Z
d
SP:{1,2} 
d2

Let  G denote the Grid which differ from G
i
only on we i.e. flip coordinate e in G.

Set
i sp(G)  SP(G)  SP( iG).
Observation
Influencee
Pr  SP(G )  SP( eG ) 
G
   i sp (G )
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.
 pr[e participates in
all the SP in G]
i sp(G)  SP(G)  SP( iG).
Proof cont.
as SP 
E  # SP  G   SP  eG 

G  e
  Influence e SP
e
2
  f S  S
S

2
 f S  S  var SP
S

And by [KKL] there is at least one variable whose
influence was as big as (n/logn)


v
var SP   f  S  S   

log
v
S


2
Mechanism Design
Shortest Path Problem
Mechanism Design Problem





N agents ,bidders, each agent i has private input
tiT. Everything else in this scenario is public
knowledge.
The output specification maps to each type vector
t= t1 …tn a set of allowed outputs oO.
Each agent i has a valuation for his items:
Vi(ti,o) = outcome for the agents.
Each agent wishes to optimize his own utility.
Objective: minimize the objective function, the
total payment.
Means: protocol between agents and auctioneer.
Truth implementation

The action of an agent consists of reporting its
type, its true type.

Playing the truth is the dominating strategy

THM: If there exists a mechanism then there
exists also a Truthful Implementation.

Proof: simulate the hypothetical implementation
based on the actions derived from the reported
types.
Vickery-Groves-Clarke (VGC)
Mechanism Design for SP
Always in the shortest
10$
path
10$
50$
50$
Shortest Path using VGC






Problem definition:
Communication network modeled by a directed
graph G and two vertices source s and target t.
Agents = edges in G
Each agent has a cost for sending a single message
on his edge denote by te.
Objective: find the shortest (cheapest) path from s
to t.
Means: protocol between agents and auctioneer.
Shortest Path using VGC



C(G) = costs along the shortest path (s,t)
in G.
compute a shortest path in the G , at cost
C(G) .
Each agent that participates in the SP
obtains the payment she demanded plus
[ C(G\e) – te ].
SP on G\e
How much will we pay?
10$
10$
50$
50$
junta

A function is a J-junta if its value
depends on only J variables.
-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1

1 -1 1
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
High vs. Low Frequencies
Def: The section of a function f above k is
f
k
 f S
S k
S
and the low-frequency portion is
f
k
 f S
S k
S
Freidgut Theorem
Thm: any Boolean f is an [, j]-junta for
O as f /  
j =2
Proof:
1.
2.
Specify the junta J
Show the complement of J has little influence
Specify the Junta
Set k=(as(f)/), and =2-(k)
Let
J  i| influencei  f    
We’ll prove:
and let
2
AJ f  1  
2
2
f'(x)  sign AJ f  x  J  
hence, J is a [,j]-junta, and |J|=2O(k)
High Frequencies Contribute Little
Prop: k >> r log r implies
f
k 2
2

2

f
S




S k
4
Proof: a character S of size larger than k
spreads w.h.p. over all parts Ih, hence
contributes to the influence of all parts.
If such characters were heavy (>/4), then
surely there would be more than j parts Ih
that fail the t independence-tests
Lemma:
Proof:
Altogether
influenceJ  f    2
influenceJ  f   f
k 2
2
+ influence
k
J

f

 
2
Altogether
influence
k
J
 f   influence  f
k
i
iJ
2


iJ iS, S k
f(S)  S
2
?
Beckner/Nelson/Bonami
Inequality
Def: let T be the following operator on any f,
E

T f  x 
Prop:
Proof:
T f 
T f  x  
z 1   / 2
f  x z 
  f S  
S
Sn 
 f S  
Sn 
S
S
 x E S z 
z
Beckner/Nelson/Bonami
Inequality
Def: let T be the following operator on any f,
T f  x 
E

z 1   / 2
f  x z 
Thm: for any p≥r and ≤((r-1)/(p-1))½
T f  f r
p
Beckner/Nelson/Bonami Corollary
Corollary 1: for any real f and 2≥r≥1
f
k
2
 r  1
k 2
fr
Corollary 2: for real f and r>2
f
k
k
r
 r  1
2
f2
Freidgut Theorem
Thm: any Boolean f is an [, j]-junta for
O as f /ε 
Proof:
1.
2.
j =2
Specify the junta J
Show the complement of J has little influence
Altogether
influence
k
J
 f   influence  f
k
i
iJ
Beckner
2


iJ iS, S k
 2O(k)  
iJ
2
O(k)
 2O(k)  
f(S)  S
iJ
2
 f(S) 
iS
iS
S
r
4/r
S
2
   influence i  f  
2/ r
iJ
 f(S) 
2
2
O(k)

as  f 
 
2
r