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Analysis of Boolean Functions CMU 18-859S, Spring 2007
PROBLEM SET 1
Due: Thursday, February 1
Homework policy : I encourage you to try to solve the problems by yourself. However, you may collaborate as long as you do the writeup yourself and list the people you talked with.
Notation used:
[n] : the set { 1 , 2 ,... , n} x(i)^ : the n-bit string x with its ith bit flipped, where i ∈ [n] S : always a subset of [n], unless otherwise specified Fn 2 : the n-dimensional vector space over the 2 -element field F 2 H⊥^ : the orthogonal complement of the subspace H of Fn 2 ; i.e., the subspace {x ∈ Fn 2 : 〈x, h〉 = 0 ∀h ∈ H} Prx, Ex, Varx : always denotes Probability, Expectation, Variance with respect to the uni- form probability distribution of x on its range, unless otherwise specified
1. Poincar´e Inequality I. Let f : {T, F}n^ → {T, F}. As in Lecture 1, define the total influence of f to be I(f ) = E x
[
#{i ∈ [n] : f (x) 6 = f (x(i))}
]
Show that 4 Pr x [f (x) = T] Pr x [f (x) = F] ≤ I(f ).
(Please give a self-contained proof.)
2. Flipping Coins. Suppose you have a biased coin which has probability p of coming up heads. You try to approximate a fair coin toss by flipping the biased coin n times and declaring “overall heads” if the number of heads you flipped was odd. Show that the probability of “overall heads” is 1 2 −^
1 2 (1^ −^2 p)
n.
3. Lagrange Interpolation. A multivariate polynomial with real coefficients is said to be multi- linear if no variable in it is raised to a power greater than 1 ; e.g., x 1 x 2 + 3x 1 x 3 x 4 −. 4 x 2 x 4 + 1. 1. In this problem we will give an alternate, direct proof (no linear algebra) that every function f : {− 1 , 1 }n^ → R can be uniquely expressed as an n-variate multilinear polynomial.
(a) Show existence by explicit construction. Use expressions like ( x 1 + 1 2
x 2 − 1 2
x 3 − 1 2
which is 1 when x = (1, − 1 , −1) and 0 elsewhere on the discrete cube.
(b) Show uniqueness by arguing that any nonzero n-variate multilinear polynomial must have a nonzero value somewhere in {− 1 , 1 }n. (Hint: Induction on n .)
4. No Weight Beyond Level 1. (a) Suppose f : {− 1 , 1 }n^ → {− 1 , 1 } satisfies ∑
|S|> 1
f^ ˆ (S)^2 = 0.
Show that f is a 1 -junta (i.e., a constant function, a dictator, or an anti-dictator).
(b) Show that the above result is not true if 1 is replaced by 2.
5. Odd Functions. A function f : {− 1 , 1 }n^ → R is said to be odd if f (−x) = −f (x) for all x ∈ {− 1 , 1 }n. Show that f is odd if and only if “f only has odd Fourier coefficients” — i.e., f^ ˆ (S) = 0 for all S of even cardinality. 6. Indicators of Subspaces. Let H be a subspace of Fn 2 of codimension d; i.e., dim(H⊥) = d. Let f : Fn 2 → { 0 , 1 } denote the indicator function of H (here 0 and 1 in f ’s range are treated as real numbers).
(a) Show that for all S ⊆ [n],
f^ ˆ (S) =
2 −d^ if S ∈ H⊥, 0 else,
where in “S ∈ H⊥” we identify the subset S with its 0 - 1 characteristic vector.
