18-859s: analysis of boolean functions. administrivia me: ryan o’donnell; email:...

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  • Slide 1
  • 18-859S: Analysis of Boolean Functions
  • Slide 2
  • Administrivia Me: Ryan ODonnell; email: [email protected] [email protected] Office hours: Wean 7121, by appointment Web site: http://www.cs.cmu.edu/~odonnell/boolean-analysis Mailing list: Please sign up! Instructions on web page. Blog: http://boolean-analysis.blogspot.com http://boolean-analysis.blogspot.com Evaluation: About 5 problem sets. 2 or 2.5 scribe notes, graded (worth equal to that of a problem set)
  • Slide 3
  • The Boolean function The boolean function
  • Slide 4
  • All things to all people
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  • x f (x) 0000 0 0001 1 0010 1 0011 1 0100 0 0101 1 0110 1 0111 1 1000 0 1001 1 1010 1 1011 1 1100 1 1101 1 1110 1 1111 1 What: Truth Table To whom: Complexity theorists, Circuit designers
  • Slide 6
  • What: Subset of the Discrete Cube To whom: Geometers of the Cube Combinatorialists, Coding theorists, Metric space types (with Hamming Distance)
  • Slide 7
  • What: Concept To whom: Machine Learning theorists Objects n features Visit our new online pharmacy store and save up to 80% From: Tami Curran To: Date: Nov 8 2006 - 12:[email protected]@cs.cmu.edu Take that! Visit our new online pharmacy store and save up to 80% Only we offer: - All popular drugs are available (Viagra, Cialis,Levitra and much much more ) - World Wide Shipping - No Doctor Visits - No Prescriptions - 100% CLICK TO FIND OUT ABOUT MORE SPECIAL OFFERS AND VISIT OUR NEW ONLINE PHARMACY STORE Viagra Cialis Levitra.com.ng Credit Mortgage Lottery ALL CAPS 1110000111100001 SPAM / NOT-SPAM message
  • Slide 8
  • What: Set System To whom: Combinatorialists, Extremal & Algebraic n element universe a set, X , a collection of subsets Set System or Hypergraph or Simplicial Complex (if f monotone)
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  • What: Graph Property To whom: Statistical physicists, Probabilists, Random k-SAT-ers an actual graph A property of graphs; eg., percolation (left-right crossing) graph with n potential edges Also good for: Ising Model Erds-Rnyi random graph model Random k-SAT satisfiability (for k-reg. hypergraphs)
  • Slide 10
  • What: Voting Scheme / Social Choice To whom: Econometricists, Political scientists 0 =1 =1 = votes winner majority electoral college dictatorship n voters
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  • What: Set of integers To whom: Number theorists, Additive combinatoricists How dense a set do you need to guarantee an arithmetic progression of length k? Suppose f indicates the primes; is there a nontrivial solution to f (x) f (x+a) f (x+2a) = 1?
  • Slide 12
  • Fourier / Harmonic Analysis of Boolean Functions = A set of techniques for studying structural properties of boolean functions.
  • Slide 13
  • What does it mean for f to be simple fair symmetric spread out / concentrated pseudo- or quasi-random low-degree ?
  • Slide 14
  • When is a Boolean function simple?
  • Slide 15
  • Juntas Definition: f : {0,1} n ! {0,1} is called an r-junta if f actually only depends on some subset of r out of n coordinates.
  • Slide 16
  • Fourier Analysis As t ! 1 : changes according to Heat Equation, a differential equation. Basic solutions: 1, sin(2 x), cos(2 x), sin(4 x), cos(4 x), sin(6 x), Every f : n ! expressible as linear combination of these frequencies. Temperature
  • Slide 17
  • Fourier Analysis of Boolean Functions Basic solutions: Parity (XOR) functions on on the 2 n subsets of coordinates Every f : {0,1} n ! expressible as linear combination of these frequencies. Fourier expansion of f, Fourier coefficients of f. Displacement? As t ! 1 : changes via a Diffusion differential equation.
  • Slide 18
  • Hallmarks of Fourier Analysis 1.Uniform probability distribution on {0,1} n. 2.Discrete cube graph structure.
  • Slide 19
  • Energy Definition: For f : {0,1} n ! {0,1}, is the average sensitivity, or edge-boundary (normalized), or total influence, or energy.
  • Slide 20
  • Energy Highest energy f ? Lowest energy f ? Lowest energy balanced f ? Majority? Random function? n/2. (Homework: f balanced ) ( f ) 1.) Parity on all bits / its negation. = n. Constants. = 0. f (x) = x i, or x i. (Dictator)
  • Slide 21
  • Connection to Circuit Complexity Theorem: [Linial-Mansour-Nisan + Hstad] If f is computable by a circuit of size S and depth D, then ( f ) O(log D1 S). In particular, f 2 AC 0 ) ( f ) polylog(n). Hence: Parity AC 0. Majority AC 0. Pseudorandom function generators AC 0.
  • Slide 22
  • Lowest Possible Energy Lowest energy balanced function that depends essentially on all n inputs? Example: (Tribes) = (log n) Friedguts Theorem: For all f : {0,1} n ! {0,1}, for all > 0, f is -close to a 2 O( ( f ) / ) -junta. log 2 n (log log n) n / log 2 n Tribes
  • Slide 23
  • When is a boolean function fair?
  • Slide 24
  • Influences Definition: The influence of the i th coordinate on f is I.e., probability i th voter is a swing voter. Proposition: ( f ) = i Inf i ( f ) x with i th bit flipped Impartial Culture (IC) assumption AKA Banzhaf Power Index.
  • Slide 25
  • Influences For a fair voting scheme, do you want influences large or small? 1Inf i (Parity) =Inf i (x j ) = Inf i (Majority n ) =Inf i (Tribes n ) = 1 if i = j, 0 else
  • Slide 26
  • Influential Coalitions Theorem: [Kahn-Kalai-Linial] If f : {0,1} n ! {0,1} is any balanced voting scheme, at least one candidate can bribe o(1) frac. of voters, win with probability 1 o(1). Corollary of: KKL Theorem: For every balanced f, there is an i with Inf i ( f ) After collecting voters, they control with probability 1 o(1). (Both theorems sharp: Tribes.)
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  • Slide 28
  • Miscounted Votes Definition: Noise sensitivity of f at flip each bit of x indep. w.p. Aside: In diffusion process:, where = exp( t).
  • Slide 29
  • The Best Scheme Against Miscounts Theorems: For all 0 , (Dictator) = (Majority n ) ! (ElectoralCollege) (Tribes n ) ! Majority Is Stablest Theorem: If f is balanced and ( f ) arccos(1 2 ) / , then Inf i ( f ) O(1/ ) for at least one i. arccos(1 2 ) / as n ! 1,
  • Slide 30
  • Applications of to P vs. NP Q: Is it possible that for every language L in NP, there is a poly-size family of circuits computing L on 100% of all inputs (of length n, for each n)? A: No. Assuming NP P (/poly). What about 99% ? What about 75% ? What about 51% ? How hard is NP on average?
  • Slide 31
  • Avg. case NP: Slightly hard ) Very hard Say f 2 NP, balanced, and slightly hard: best poly-size circuit is 99% right. Impagliazzos Hard Core Theorem: 9 H {0,1} n of size 2% 2 n such that no poly-sized circuit can compute f on ( + negl.) fraction of H. On typical input to F, about 2% 10 6 of the f -inputs come from H. 2% (Tribes 10 6 ) 49% Tribes fffff n 10 6 Let F : {0,1} 10 6 n ! {0,1} be 2 NP. (Why?) Theorem: F is not 51%-computable by poly-circuits. )
  • Slide 32
  • When is a boolean function pseudo- or quasi-random?
  • Slide 33
  • The Opposite of Pseudorandom Given f s value on M random points, can you predict f at other points? One idea: Take some weighted majority of known f-values, based on Hamming distance. Can this work with M 2 n ? 010100111 110101111 001010111 010100100 001010010 111011111 f (x i )xixi Examples: Predict: f ( 00010101 ) = ? Learning f (from random examples)
  • Slide 34
  • Learning from Random Examples Works if f has long-range correlations e.g., small ( f ) or ( f ). LMN Algorithm: This will work (using an appropriate weighted majority) if M n O( ( f )). E.g., depth-D, poly-size circuits predictable after only examples. Similar theorem exists for functions with small .
  • Slide 35
  • Learning with Queries Goldreich-Levin Theorem: From any one-way function g : {0,1} n ! {0,1} n, can produce a hard-core predicate f : {0,1} 2n ! {0,1}. Proof by contraposition: gives a learning algorithm, using queries, for learning f s large Fourier coefficients. GL algorithm put to positive use in Learning Theory: Theorem: [Mansour] Poly-size DNF (depth-2 circuits) learnable with queries in time n O(log log n) ; Fourier techniques. Jacksons Theorem: Improved to poly time & queries, by adding an ML technique.
  • Slide 36
  • Quasirandomness Fix a small set of simple statistical tests; quasirandom if you pass all of them. For graphs: Graph G with edge density p is quasirandom if, for each O(1)-size graph H, G has the roughly the expected number of copies of H. For boolean functions:Function f with E[ f ] = p is quasirandom if, (one weak possible notion) for each O(1)-junta h : {0,1} n ! {0,1}, f has roughly 0 correlation with h. (I.e., given h(x), youd still guess p for Pr[ f(x) = 1].)
  • Slide 37
  • Quasirandomness & Tests Hstads Test: Pick x ~ {0,1} n uniformly. Pick y ~ {0,1} n uniformly. Set z = x y. Set w ~ z. Test whether f (x) f (y) f (w) = 0. f balanced and random: would pass with probability . f a Dictator: would pass with probability 1 . Theorem: If f is balanced and quasirandom, passes test with probability + o (1). Almost the canonical Fourier Analysis problem; where well start. x = 101000011011111 y = 000100000101110 z = 101100011110001 w = 001100010100000
  • Slide 38
  • Hstads Hardness of Approximation Corollary: [Hstads Test + PCP machinery] Given a system of 3-variable linear equations mod 2, x 1 x 3 x 7 = 0 x 2 x 4 x 7 = 1 x 1 x 5 x 6 = 0 x 6 x 8 x 9 = 0 which is 99%-satisfiable, no efficient algorithm can find a solution satisfying 51% of equations. (Unless P = NP.) e.g.,
  • Slide 39
  • Proof Idea Test yields an NP-hardness gadget for reductions, m-coloring graphs 3-variable mod-2 equations. Blocks are 99%-satisfiable because of Dictators these encode the m colors. Hstads Test Theorem: any f satisfying 51% of a block is noticeably correlated with O(1) coordinates ) decodable to O(1) Dictators/colors. prob.05 of testing: f (000) f (010) f (011) = 0 weight.05: x 000 x 010 x 011 = 0 vertex block of 2 m vars, x 00 0, x 00 1,
  • Slide 40
  • Thursday: When is a boolean function linear? And what is its Fourier expansion?