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Analysis of Boolean Functions and Complexity Theory Economics Combinatorics Etc. Slides prepared with help of Ricky Rozen. Influential People. - PowerPoint PPT Presentation
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Analysis of Boolean Analysis of Boolean FunctionsFunctions
andandComplexity TheoryComplexity Theory
EconomicsEconomicsCombinatoricsCombinatorics
Etc.Etc.
Slides prepared with help of Ricky Slides prepared with help of Ricky RozenRozen
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 according not necessarily according to majority…to 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
Noise SensitivityNoise 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 outIt turns out: one cannot design : one cannot design an an ff that would be robust to that would be robust to such noise --that is, would, on such noise --that is, would, on average, change value w.p. average, change value w.p. < < O(1)O(1)-- unless determining the -- unless determining the outcome according to very few outcome according to very few of the votersof the voters
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
1 / 2 12iinfl uence
n
n
nO
n
1
1 / 2 12iinfl uence
n
n
nO
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
Average SensitivityAverage 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) = n(Majority) = 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
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
Distribution over CharactersDistribution over Characters
SimpleSimple ObservationsObservations
ClaimClaim::
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
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 linear is 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
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
First Passage Percolation First Passage Percolation [BKS][BKS]
Each edge costs a w/probability ½ and b w/probability
½
First Passage PercolationFirst Passage Percolation
Consider the GridConsider the Grid
For each edge For each edge ee of choose of choose independentlyindependently wwee = 1 = 1 or or wwee = 2 = 2, each with probability , each with probability ½½
This induces a shortest-path metric onThis induces a shortest-path metric on
ThmThm : The variance of the shortest path : The variance of the shortest path from the origin to vertex from the origin to vertex vv is bounded is bounded from above by from above by O( |v|/ log |v|) O( |v|/ log |v|) [BKS][BKS]
Proof ideaProof idea: The average sensitivity of : The average sensitivity of shortest-path is bounded by that termshortest-path is bounded by that term
dZdZ
dZdZ
dZdZ
LetLet GG denote the griddenote the grid
SPSPGG – the shortest path in – the shortest path in GG from the from the origin to origin to vv..
Let denote the Grid which differ Let denote the Grid which differ from from GG only on only on wwee i.e. flip the value of i.e. flip the value of ee in in GG..
Set Set
dZ
Proof outlineProof outline
2dSP:{1,2}
.( ) ( ) ( )i isp G SP G SP G
iG
ObservationObservation
e eG
I nfl uence Pr SP(G) SP(σ G)
pr[e participates in
all the S
=
P in G]
e eG
I nfl uence Pr SP(G) SP(σ G)
pr[e participates in
all the S
=
P 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.
Proof cont.Proof cont.
And by [KKL] there is at least one variable And by [KKL] there is at least one variable whose influence is at least whose influence is at least (logn/n) (logn/n)
eeG
2
S
as SP E # SP G SP G
f S S var SP
eeG
2
S
as SP E # SP G SP G
f S S var SP
2
S
vvar SP f S S
log v
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.
In other words:In other words: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
(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..
Example-Max CliqueExample-Max Clique
Consider Consider GGG(n,p)G(n,p)
The size of the interval of The size of the interval of probabilities probabilities pp for which the clique for which the clique number of number of GG is almost surely is almost surely kk (where (where k k log log nn) is of order ) is of order loglog-1-1nn..
The threshold interval: The transition The threshold interval: The transition between clique numbers between clique numbers k-1k-1 and and kk..
Probability for choosing an edge
Number of vertices
The probability of having a (The probability of having a (k k + 1+ 1)-clique )-clique is still small (is still small ( log log-1-1nn). ).
The value of The value of pp must increase bymust increase by clogclog-1-1nn 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
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
ThmThm [Margulis-Russo]: [Margulis-Russo]:
For monotoneFor monotone ff
HenceHenceLemmaLemma::For monotoneFor monotone ff > 0 > 0, , q q[p, p+[p, p+]] s.t. s.t. asasqq(f) (f) 1/ 1/
ProofProof:: Otherwise Otherwise p+p+(f) > 1(f) > 1
d (f )(f )
dq
as q
q
d (f )(f )
dq
as
ProofProof [Margulis-Russo]: [Margulis-Russo]:
i
n nq q q
i qi 1 i 1i
d fff (f )
dq q
influence as i
n nq q q
i qi 1 i 1i
d fff (f )
dq q
influence as
Mechanism Design Mechanism Design ProblemProblem
NN agentsagents, each agent , each agent ii has has privateprivate input input ttiiTT. . All other information isAll other information is publicpublic knowledge.knowledge.
Each agent Each agent ii has a has a valuationvaluation for all items: for all items: Each agent wishes to optimize her own utility.Each agent wishes to optimize her own utility.
ObjectiveObjective: minimize : minimize the objective function, the objective function, the total payment.total payment.
MeansMeans: protocol between agents and : protocol between agents and auctioneerauctioneer..
Vickery-Clarke-Groves Vickery-Clarke-Groves (VCG)(VCG)
Sealed bid auctionSealed bid auction A A Truth RevealingTruth Revealing protocol, namely, protocol, namely,
one in which each agent might as one in which each agent might as well reveal her valuation to the well reveal her valuation to the auctioneerauctioneer
Whereby each agent gets the best Whereby each agent gets the best (for her) price she could have bid and (for her) price she could have bid and still win the auction still win the auction
Shortest Path using VGCShortest Path using VGC
Problem definition:Problem definition: Communication networkCommunication network modeled by a directed modeled by a directed
graph graph GG and two vertices source and two vertices source ss and target and target tt.. AgentsAgents = edges in = edges in GG Each agent has a cost for sending a single Each agent has a cost for sending a single
message on her edge denote by message on her edge denote by ttee..
ObjectiveObjective:: find the shortest (cheapest) path find the shortest (cheapest) path from from ss to to tt..
MeansMeans:: protocol between agents and protocol between agents and auctioneer.auctioneer.
VCG for Shortest-PathVCG for Shortest-Path
50$
10$
50$
10$Always in the shortest
path
How much will we pay?How much will we pay?
SPSP
Every agent will get 1$ more.Every agent will get 1$ more.
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$2$
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
[n][n]x
IIz
[n][n]
Noise-SensitivityNoise-Sensitivity
How often does the value of How often does the value of ff changes changes when the input is perturbed?when the input is perturbed?
x
IIz
DefDef((,p,x,p,x[n] [n] ): Let ): Let 0<0<<1<1, and , and xxP([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~
[n][n] is a is a noise subsetnoise subset, and, and z~ z~ pp
II is a is a replacementreplacement..
DefDef((--noise-sensitivitynoise-sensitivity): let ): let 0<0<<1<1, then, then
[ When [ When p=½p=½ equivalent to flipping each equivalent to flipping each coordinate in coordinate in xx independently w.p. independently w.p. /2/2.].]
[n] [n]p ,p,xx~ ,y~
ns f = Pr f x f y
[n] [n]p ,p,xx~ ,y~
ns f = Pr f x f y
[n][n]xIIz
Noise-SensitivityNoise-Sensitivity
Noise-Sensitivity – Cont.Noise-Sensitivity – Cont.
AdvantageAdvantage: very efficiently testable (using : very efficiently testable (using only two queries) by a only two queries) by a perturbation-testperturbation-test..
DefDef ((perturbation-testperturbation-test): choose ): choose x~x~pp, and , and y~y~,p,x,p,x, 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 ff..
PropProp: the : the -noise-sensitivity is given by -noise-sensitivity is given by
2S
S
2 ns f =1 1 f S 2S
S
2 ns f =1 1 f S
Relation between Relation between ParametersParameters
PropProp: small : small nsns small small high-freq weighthigh-freq weight
ProofProof::therefore: therefore: if if nsns is small, then is small, then Hence the Hence the high frequencieshigh frequencies must must have small weights (ashave small weights (as ). ).
PropProp: small : small asas small small high-freq weighthigh-freq weight
ProofProof:: 2
S
ff (S) S as 2
S
ff (S) S as
2S
S
2 ns f =1 1 f S 2S
S
2 ns f =1 1 f S
2S
S
1 f S ~1 2S
S
1 f S ~1
2
S
f S 1 2
S
f S 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
kS
S k
ff S
k
SS k
ff S
kS
S k
ff S
k
SS k
ff S
Low-degree B.f are Juntas Low-degree B.f are Juntas [KS][KS]
TheoremTheorem: :
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(-2-2kk332k2k))
CorollaryCorollary: :
fix a fix a pp-biased distribution -biased distribution pp over over P([n])P([n])
Let Let >0>0 be any parameter. be any parameter.
Set Set k=logk=log1-1-(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(-2-2kk332k2k))
2k22
f Ok
2k22
f Ok
2ns f O k 2ns f O k
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
Long-CodeLong-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 dictatorshipsmonotone 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 nn elements elements
Long-CodeLong-Code
Encoding an element Encoding an element ee[n][n] :: EEee legally-encodeslegally-encodes an element an element ee if if EEee = f = fee
FF FF TT TT TT
Codes and Boolean Codes and Boolean FunctionsFunctions
DefDef: an : an mm-bit code is a subset of the set of -bit code is a subset of the set of all the all the mm-binary string -binary string
CC{-1,1}{-1,1}mm
The The distancedistance of a code of a code CC is the minimum, is the minimum, over all pairs of legal-words (in over all pairs of legal-words (in CC), of the ), of the Hamming distance between the two wordsHamming distance between the two words
NoteNote: A Boolean function over : A Boolean function over nn binary binary variables is a variables is a 22nn-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 22nn-bits code-bits code
Long-Code Long-Code Monotone- Monotone-DictatorshipDictatorship
In the long-code, the legal code-In the long-code, the legal code-words are all monotone dictatorshipswords are all monotone dictatorships
C={C={{i}{i} | i | i [n]} [n]}
namely, all the singleton charactersnamely, all the singleton characters
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 according not necessarily according to majority…to majority…
And what ifAnd what ifsomeonesomeone(in Florida?)(in Florida?)can flipcan flipsome votessome votes
PowerPower
influenceinfluence
Of course they’ll have to discuss it
over dinner….
Open QuestionsOpen Questions
Mechanism DesignMechanism Design: show a non truth-: show a non truth-revealing protocol in which the pay is revealing protocol in which the pay is smaller (Nash equilibrium when all agents smaller (Nash equilibrium when all agents tell the truth?)tell the truth?)
Hardness of ApproximationHardness of Approximation:: MAX-CUTMAX-CUT ColoringColoring a 3-colorable graph with fewest colors a 3-colorable graph with fewest colors
Graph PropertiesGraph Properties: find sharp-thresholds for : find sharp-thresholds for propertiesproperties
AnalysisAnalysis: show weakest condition for a : show weakest condition for a function to be a Juntafunction to be a Junta
Apply Apply Concentration of MeasureConcentration of Measure techniques techniques to other problems in Complexity Theoryto other problems in Complexity Theory
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
Hadamard CodeHadamard Code
In the Hadamard code theIn the Hadamard code theset of legal-words consists of set of legal-words consists of all multiplicative (linear if all multiplicative (linear if over over {0,1}{0,1}) functions) functions
C={C={SS | S | S [n]} [n]}
namely all characters namely all characters
6262
Hadamard Test – SoundnessHadamard Test – Soundness
PropProp(soundness):(soundness):
ProofProof::
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 SS ,S ,S
1 2 3 x,y S S S SS ,S ,S
1 2 3 x S S y S SS ,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-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.
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
Perturbation Perturbation
DefDef: denote by : denote by the distribution the distribution over all subsets of over all subsets of [n][n], which , which assigns probability to a subset assigns probability to a subset xx as follows:as follows:
independently, for each independently, for each ii[n][n], let, let iixx with probability with probability 1-1- iixx with probability with probability
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-
Long-code Tests Long-code Tests
Def Def (a (a long-code testlong-code test): given a code-): given a code-word word ww, probe it in a constant , probe it in a constant number of entries, andnumber of entries, and accept w.h.p if accept w.h.p if ww is a monotone is a monotone
dictatorshipdictatorship reject w.h.p if reject w.h.p if ww is not close to any is not close to any
monotone dictatorshipmonotone dictatorship
Efficient Long-code TestsEfficient Long-code Tests
For some applications, it suffices if the test For some applications, it suffices if the test may accept illegal code-words, nevertheless, may accept illegal code-words, nevertheless, ones which have short list-decoding:ones which have short list-decoding:
DefDef(a (a long-code list-testlong-code list-test): given a code-word ): given a code-word ww, 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 ww is not evenis not even approximately approximately
determined by a short list of domain elementsdetermined by a short list of domain elements, , that is, if that is, if a a JuntaJunta JJ[n][n] s.t. s.t. f f is close to is close to f’ f’ and and f’(x)=f’(xf’(x)=f’(xJ) J) for allfor all x x
NoteNote: a long-code list-test, distinguishes : a long-code list-test, distinguishes between the case between the case ww is a is a dictatorshipdictatorship, to the , to the case case ww is far from a is far from a juntajunta..