TIN 085: Selected Topics in Computational Complexity I
Assignment 1
Deadline: November 28, 2016
General Instructions:
• Write your solutions by furnishing all relevant details (you may assume the results already
covered in the class).
• You are strongly encouraged to solve the problems by yourself. You may discuss but write
the solutions on your own. Any copying will get zero in the whole assignment.
• Please submit the answers in soft-copy form (as a single pdf file) via email to [email protected].
A function f : {−1, 1}n → {−1, 1} is a dictator function if there exists an input index i ∈ [n] such
that f (x) = xi .
Observe that dictator functions are basically the functions χS such that S ⊆ [n] and |S| = 1.
So these functions form a subclass of the class of linear functions. In the class we have seen a test
to verify whether a given function is linear or far from being linear. However you may check that
that test fails to verify whether the given function is dictator or far from being dictator.
Let us consider the following testing algorithm:
1. Pick up x, y ∈ {−1, 1}n uniformly and independently;
2. Pick up w = (w1 , · · · , wn ) ∈ {−1, 1}n by independently choosing each wi to be −1 with
probability δ;
3. Accept if and only if f (x · y · w) = f (x)f (y), where the operation ” · ” denotes pointwise
multiplication described in the class.
Question 1: [3+5]
• When input is chosen uniformly from {−1, 1}n , what will be the expected value of the function
χS for S ⊆ [n]?
• When input is chosen according to the distribution that is used to pick w in the step 2 of the
above described test, what will be the expected value of the function χS for S ⊆ [n]?
Now recall how we calculated the probability of acceptance of BLR Test in the class and use
similar kind of technique to show the following.
1
Question 2: [10] For any function f : {−1, 1}n → {−1, 1}, the above described test will
accept with probability
1 1 X
(1 − 2δ)|S| (fˆ(S))3 .
+
2 2
S⊆[n]
Question 3: [2] Now use the above to show that if the given function is a dictator then the
test will accept with probability at least (1 − δ).
Again use Question 2 to show the following.
Question 4: [10] If the test accepts with probability at least 1 − 4δ
3 , then the given function
f is 4δ
-close
to
either
a
dictator
function
or
the
function
that
always
ouputs
one.
3
Question 5: [10] Try to augment the above test to ensure that the final test does not accept
if the given function f is 4δ
3 -close to the function that always ouputs one. How many extra queries
to the function f do you need for that purpose? How will it affect the acceptance probability?
2