Lecture 23: Analysis of Boolean Functions

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1 CCI-B609: A Theorist s Toolkit, Fall 016 Nov 9 Lecture 3: Analysis of Boolean Functions Lecturer: Yuan Zhou cribe: Haoyu Zhang 1 Introduction In this lecture, we will talk about boolean function f : {0, 1} n {0, 1} that maps n bits into a single bit. We will find that the boolean function is an abstract of all functions we use in computer science, since all computers are implemented by circuits which have on/off(0/1 two states. For our convenience, we use another represent of f, which is f : { 1, 1} n { 1, 1}, and in some more general cases f : { 1, 1} n R. Here are eamples of boolean functions. Majority function Majority function over 3 bits is defined as: maj 3 ( 1,, 3 = sgn( = { 1 if < 0 1 otherwise. Parity function Parity function over 3 bits is defined as: parity( 1,, 3 = 1 3. Note that, we could rewrite the majority function over 3 bits as maj 3 ( 1,, 3 = 1 ( ( ( ( ( 1 1 ( ( 1 1 (

2 Lecture 3: Analysis of Boolean Functions Epanding it out, we have maj 3 ( 1,, 3 = This is also called interpolation. In fact, we could epress all f : { 1, 1} n { 1, 1} in this way, which we will discuss later. Fourier Analysis for boolean functions We first define several notations we will use later. Definition 1. For each subset [n, let χ ( = i. i Definition. Inner product f, g { 1, 1} n R, let f, g = E f(g(. { 1,1} n Then, we will have the following lemma. Lemma 1. For, T [n, we have χ (, χ T ( = Proof. By our definition of inner product, { 0 if T 1 otherwise. χ (, χ T ( χ (χ T ( { 1,1} [ n i i i i T i [ i i T i T i T i, where T denotes their symmetric difference.

3 Lecture 3: Analysis of Boolean Functions 3 When = T, T =, so χ (, χ T ( = 1. When T, j T, such that E [ i T i [ i T,i j i E j [ j = 0. By the lemma, we know that χ ( for all [n are orthonormal basis, and we could represent any boolean function with the basis. Fact 1. All functions f { 1, 1} n R can be epressed in the following form uniquely f = [n f(χ (, where f( = f, χ (. The epression of f is called Fourier transform of f, and f( is called inner Fourier transform. With the Fourier transform, we could proof the following lemma. Lemma. (Parseval Identity For all, T [n, we have f( [f( = 1. [n Proof. E[f( f(χ ( f(t χ T ( [n T [n f( f(t χ (χ T ( =,T [n,t [n = f(. [n f( f(t E [χ (χ T (

4 Inf i (f = 1 [1 + i = 1 [ Lecture 3: Analysis of Boolean Functions 4 Net, we talk about a new concept. Definition 3. The influence of the variable i on the function f : { 1, 1} n { 1, 1} is Inf i (f = P r [f( f( i, where the i denotes the vector after a value change on the i-th bit of. Here are some eamples: Inf i (maj = Θ( 1 n, Inf i (parity = 1. Using the Fourier transform, we could epress the influence function into analytical form. Lemma 3. Inf i (f =, i f( Proof. 1 f(f( i Inf i (f = 1 1 E [f(f( i. Using the Fourier transform, we have E[f(f( i f(χ ( f(t χ T ( i [n T [n = f( f(t [ E χ (χ T ( i,t [n = f( + f(. i i Then we finally get f( i f( f( + i f( i f( =, i f(. Definition 4. Average sensitivity A(f = avginf i (f = [n n f(.

5 =,T,U Lecture 3: Analysis of Boolean Functions 5 3 Application: Property testing Now we consider an application of the Fourier transform of boolean functions. The property testing problem is, given a boolean function f : { 1, 1} n { 1, 1}, we want to test if it has a property P. Our goal is: find an algorithm, given f, the algorithm make q queries of f. If f has property P, the algorithm outputs accept with probability 1(completeness; the algorithm outputs reject with probability δ > 0 if f is ɛ-far from all g P (soundness. Here we define function f is ɛ-far from g, if dist(f, g = { : f( g(} n > ɛ. More specifically, we consider the problem of linearity testing, where P = {χ [n}. The algorithm in [BLR90 for linearity testing is: 1. Generate, y { 1, 1} n randomly.. If f(f(yf(y = 1, return accept. 3. Otherwise, return reject. We have 3 queries in this algorithm. The completeness of the algorithm is obvious: if f = χ, then χ (χ (yχ (y = 1. For the soundness, we have the following claim. Claim 1. If f is ɛ-far from all g P, Pr[ reject ɛ using algorithm in [BLR90 for linearity testing. Proof. Pr[ reject = Pr[f(f(yf(y = 1 1 f(f(yf(y,y = 1 1 E [f(f(yf(y,,y where [ E [f(f(yf(y,y,y,y [,T,U f(χ ( f(t χ T (y f(uχ U (y T U f( f(t f(uχ (χ T (yχ U (χ U (y f( f(t f(u E [χ (χ U ( E y [χ T (yχ U (y = f( 3.

6 Lecture 3: Analysis of Boolean Functions 6 If f is ɛ-far from all g P, assume Pr[ reject < ɛ, then we have ɛ > Pr[ reject = 1 1 f( 3, which indicates f( 3 > 1 ɛ. On the other hand, f( 3 f( ma { f(} = ma { f(}. Then we know that, =, such that f( > 1 ɛ. If we consider χ, 1 dist(f, f(χ = 1 1 f, χ = 1 1 f( < ɛ. This contradicts with the f is ɛ-far from all g P. Note that, we could boost the probability to high probability by repeating the algorithm O( 1 ɛ times. References [BLR90 Blum M, Luby M, Rubinfeld R. elf-testing/correcting with applications to numerical problems. Proceedings of the twenty-second annual ACM symposium on Theory of computing (TOC, 1990:

Lecture 8: Linearity and Assignment Testing

CE 533: The PCP Theorem and Hardness of Approximation (Autumn 2005) Lecture 8: Linearity and Assignment Testing 26 October 2005 Lecturer: Venkat Guruswami cribe: Paul Pham & Venkat Guruswami 1 Recap In