CPSC 468/568 (Fall 2012) HW6 Solution Set
17.1 Bayes Net
Given a 3CNF formula x, let xi = x|(x1 = x2 = . = xi = 1) (i.e.
substitute 1 for all x1 , ., xi and simplify into another 3CNF). Let f (x) be
the percentage of satisfying assignments of x where x1 =
CPSC 468/568 (Fall 2012) HW3 Solution Set
5.7 APSPACE = EXP
() Suppose L AP SP ACE , and let M be a poly-space ATM deciding L.
We build a TM M that decides L by computing the conguration graph of
M on a given input x, and then marking nodes with ACCEPT as
CPSC 468/568 Exam
April 26, 2007
Answer five of the following six questions. If you answer all six, the first five of your
answers will be graded, and the sixth will be ignored. Please remember to write your name,
CPSC 468/568, and todays date on the cove
Solution Set for CPSC 468/568 Exam 2
Question 1
(a) Unknown. We know that PH P#P PSPACE (the first inclusion is Todas Theorem)
but not whether these inclusions are proper.
(b) Unknown. We know that BPP
p
2
p
2
(see Theorem 7.18 in Arora-Barak) but
not wh
CPSC 468/568 Exam
March 1, 2007
Answer five of the following six questions. If you answer all six, the first five of your
answers will be graded, and the sixth will be ignored. Please remember to write your name,
CPSC 468/568, and todays date on the cover
Solution Set for CPSC 468/568 Exam 1
Question 1
(a) No, A is decidable. To decide whether (, x) is in A, simulate M on input x for t=|x|3
steps, output 1 if the simulation has halted by the tth step, and output 0 otherwise.
(b) Turing Machine M is oblivio
CPSC 468/568 (Fall 2012) HW5 Solution Set
7.8 3SAT BP N P P H = p
3
In problem 7.7, we showed that BP N P N P/poly . We can also use
the idea that p p P H = p . So it suces for us to prove that
3
3
3
3SAT N P/poly p p .
3
3
We will accomplish this by show
CPSC 468/568 (Fall 2012) HW4 Solution Set
7.7 BP N P N P/poly
Suppose L BP N P , and let M be a poly-time PTM giving a randomized reduction from L to 3SAT . Then for all x, P rr [L(x) = 3SAT (M (x, r)]
2/3, where r is uniformly distributed over cfw_0, 1m
CPSC 468/568 (Fall 2012) HW2 Solution Set
Problem 3.3
Acknowledgement: This solution was provided by David Costanzo, who
served as the TA for CPSC 468/568 several times, most recently in 2010.
We use essentially the same denition of B as in the textbooks
CPSC 468/568 (Fall 2012) HW1 Solution Set
1.2 TM Language Conversion
Let M = (, Q, ) and M = (, Q, ), and suppose M has k tapes. We
want to insert states and transitions into M so that it performs a cycle of
three multi-step tasks: read, write, and move
c) (7 points) Dene a linearization operator LXi on polynomials h Zp [X1 , X2 , . . . , Xn ], and
briey explain the role of linearization in the proof that IP = PSPACE.
Question 4: Probabilistic reductions and Todas Theorem (20 points):
Recall the followin
Answer Key for Exam 2 in CPSC 468/568
(December 6, 2012)
Answer to Question 1:
Justications are presented for educational purposes only. They were not required for full
credit.
a) True. A checker is just a particular type of oracle proof system.
b) Unknow