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 today’s date on the covers of all blue books you submit.
Question 1
State whether each of the following claims is true, false, or unknown. If you answer true or
false, give a very brief justification.
(a) (5 points) P#P = PSPACE
(b) (5 points) BPP ⊆ NP ∪ coNP
(c) (5 points) coNP ⊆ IP
(d) (5 points) There is a three-round Arthur-Merlin (aka “public-coin”) proof system for
Graph Non-Isomorphism. (Here, a “round” is a move by Arthur followed by a move by
Merlin.)
Question 2
(a) (5 points) Recall that Fortnow, Rompel, and Sipser [FRS] proved that MIP, the class of
languages accepted by multi-prover, interactive proof systems, is equivalent to the class
of languages accepted by probabilistic, polynomial-time (ppt) oracle Turing Machines.
State precisely the [FRS] definition of “L is accepted by a ppt oracle Turing Machine M.”
(b) (5 points) In MIP, an “oracle” convinces a ppt verifier that a string is in a language. In
IP, a “prover” convinces a ppt verifier that a string is in a language. Briefly explain the
essential difference between an oracle and a prover.
(c) (10 points) Recall that Graph Non-Isomorphism is in IP; a fortiori, it is in MIP. Prove
that Graph Non-Isomorphism is in MIP by giving a ppt oracle Turing Machine that
accepts it.
Question 3
(a) (9 points) For three points each, define the complexity classes RP, coRP, and BPP.
(b) (11 points) Recall that L is in the class ZTIME(T(n)) if there is a probabilistic Turing
Machine M that has expected running time O(T(n)) and never errs. That is, for all x in L,
M accepts x with probability 1; for all x not in L, M halts with probability 1 without
accepting x; and, for all x, the expected running time (where expectation is computed
over the coin tosses of M on input x) is O(T(|x|)). ZPP is the union, over all polynomials
p, of ZTIME(p(n)).
Prove that ZPP = RP ∩ coRP.
