Theoretical Computer Science, 2016-2017
Intermediate exam. Time limit: 180 minutes.
18/2/2017
This exam contains 2 pages and 5 questions. The total of all points is 18. Your grade for the
final exam is the minimum of the number of points you earned and 16. You have a laptop at your
disposal with all PDF-files of books and lecture notes that have been used in the class. You can
make references to any of these materials. You can also make references to claims in exercises.
You are not allowed to use the laptop for any other purpose.
1. Definition: let A be a language and X be a complexity class. The class XA is the set of
languages that are decidable by the machines that satisfy the same conditions as in the
definition of S
X but additionally have oracle access to A. Let Y also be a complexity class.
Y
Then X = A∈Y XA .
(4)
Are the following statements true or false? Prove or disprove.
(a) If a language can be decided by a Turing machine that does not read or write more than
one cell at the right of its input, then the language is regular.
(b) SPACE(n3 ) ⊆ NSPACE(n).
(c) BPPBPP = BPP.
(d) If n = pe for some prime number p and some e ≥ 2, then n is not a Carmichael number.
(e) If PSAT = NP then NP = coNP.
(f) The following set is PSPACE-complete:
nD
E
o
∃x1 ∀x2 . . . ∃x2n−1 ∀x2n Φ(x1 , . . . , x2n ) : Φ is a monotone Boolean formula
A Boolean function f : {0, 1}n → {0, 1} is monotone if f (u0v) ≤ f (u1v) for all paris
(u, v) of strings such that |uv| = n − 1. In this definition the formula φ is encoded by
OR, AND and negation operations.
(g) If A ∈ IP then the complement of A is also in IP.
(h) If a language A ∈ IP has an interactive proof system such that the prover sends at most
O(log n) bits, then the language is in BPP.
2. Suppose we generate a random language B in the following way: for every n, with probability
1/2, B has no strings of length n, and with probability 1/2 we pick a random string to be in
B. Show that for each ε > 0 we have that PB 6= NPB with probability at least 1 − ε. Hint:
modify one of the proofs of Chapter 9.
(4)
3. In this exercise you are asked to adapt the argument that #SAT is in IP. A black box
contains a polynomial p in a field of exponential size over n variables. The degree in each
variable is bounded by n. This polynomial is a very complex object and any description of it
has exponential length. On input x1 , . . . , xn , this black box evaluates this polynomial.
(4)
A verifier wants to check n values of the polynomial:
p(u1 , . . . , un ) = a
p(v1 , . . . , vn ) = b
...
p(z1 , . . . , zn ) = e,
Theoretical Computer Science Intermediate exam - Page 2 of 2
18/2/2017
but he can use the black box at most once. Fortunately, a prover claims to know p and
has unlimited computational power to convince the verifier the values are correct. Is there a
protocol such that a prover can convince the verifier in polynomial time that the values are
correct?
Note that the following does not work: the prover sends the polynomial, then the verifier
checks all values, and finally he checks whether the sent polynomial p is correct by sampling
the device on a single random point. This would work if p had a description of polynomial
size, but this is not the case.
4. For those who choose property testing. We say that an image represents a rectangle if
it is possible to draw a rectangle that contains precisely all black pixels. Thus the pictures
that contain only white or black pixels also represent rectangles. Show that there exists a
tester that runs in time O(1/ε).
(4)
5. For those who choose statistical learning theory. For a polynomial p, let fp be the classifier
(
1 if p(x) > 0
fp (x) =
0 otherwise.
(2)
Let T Pd1 be the set of all fp for real polynomials of degree at most d in one variable. Show
that VC-dimension of T Pd1 equals d + 11. In a similar way we can define fp for polynomials
with two variables. Let T Pd2 be the set of all such fp that have degree at most d in each
variable (thus for a fixed value x0 , the polynomial y → p(x0 , y) has at most degree d, and
similar for a fixed value y0 of the second argument). Show that the VC-dimension of T Pd2 is
at least (d + 1)2 .