Analytical CombinatoricsAnalytical Combinatorics

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 every variables The values of every variables may, independently, change may, independently, change with probability with probability

It turns outIt turns out: no Boolean : no Boolean ff is is robust under noise --that is, robust under noise --that is, would, on average, change w.p. would, on average, change w.p. <sqrt(<sqrt())-- unless the outcome is -- unless the outcome is almost always determined by almost always determined by very few variables very few variables (disregarding all but (disregarding all but exp(1/ exp(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

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

XORXOR : : {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 (Average Total-Influence (Average Sensitivity)Sensitivity)

DefDef: : thethe Total-InfluenceTotal-Influence of of ff ((asas) is the ) is the sum of influences of all variables sum of influences of all variables ii[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

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

S

S

ff S S

S

ff S

NormsNormsDefDef:: ( (ExpectationExpectation) norm on the function) norm on the function

ThmThm [Parseval]: [Parseval]:

for a Boolean for a Boolean ff

q q

q x P[n]ff (x)

q q

q x P[n]ff (x)

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

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

Heuristics: Hardness Heuristics: Hardness AmplificationAmplification

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

Monotone Substitute for Monotone Substitute for XORXOR

ClaimClaim::for monotone functions for monotone functions I[f] < sqrt nI[f] < sqrt n

Find a monotone function Find a monotone function ff so that almost so that almost all input settings x have sqrt n pivotal bitsall input settings x have sqrt n pivotal bits

PercolationPercolation

Each edge occurs w/probability ½

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

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 Thm [[Goldreich-Levin, Kushilevitz-Goldreich-Levin, Kushilevitz-MansourMansour]]: : f:{0,1}f:{0,1}nn{0,1}{0,1} concentrated is learnableconcentrated is learnable

Thm Thm [Akavia, Goldwasser, S.][Akavia, Goldwasser, S.]: over : 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]S]: Juntas are : 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 that ff changes its value when redrawing a subset of changes its value when redrawing a subset of its variables according to the its 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

Redraw each value of the subset, Iwith probability p

Redraw 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]: Stable B.f. are Juntas: Stable B.f. are JuntasThmThm [MOO][MOO]: Majority Stablest if low Inluence: Majority Stablest if low Inluenceii

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

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

Redraw 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

Coding TheoryCoding Theory

DefDef: a binary code is : a binary code is C C {-1, 1} {-1, 1}tt RateRate: : log|C|/tlog|C|/t DistanceDistance: : DD such that for any such that for any x, yx, yCC

H(x, y) ≥ DH(x, y) ≥ D

A string of lengthA string of length 2 2nn is a Boolean is a Boolean functionfunction {-1, 1} {-1, 1}nn {-1, 1} {-1, 1}, hence a , hence a code is a class of Boolean functionscode is a class of Boolean functions

Hadamard codeHadamard code: : all charactersall characters Long CodeLong Code: : all dictatorshipsall dictatorships

Testing Codes (PCP related)Testing Codes (PCP related)

Def Def (a (a code list-testcode list-test): given an ): given an ff, , probe it in a constant number of probe it in a constant number of entries, andentries, and accept (almost) always if accept (almost) always if f f is legalis legal reject w.h.p if reject w.h.p if ff does not havedoes not have a a

positive correlation with any legal positive correlation with any legal code-word code-word

If not rejected, there is a short If not rejected, there is a short list of legal code-words with list of legal code-words with positive correlationpositive correlation

Hadamard TestHadamard Test

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

f(x)f(y)=f(xy)f(x)f(y)=f(xy)

PropProp(completeness): a legal (completeness): a legal Hadamard word (a Hadamard word (a character) always passes character) always passes this testthis test

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-

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.

Open QuestionsOpen Questions

Entropy ConjectureEntropy Conjecture [FK] [FK] Classify functions that are closed under a large Classify functions that are closed under a large

subgroup of subgroup of SSnn

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

Graph PropertiesGraph Properties: find real sharp-thresholds for : find real 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?)

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