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 today’s date on the covers of all blue books you submit.
Question 1
(a) (2 points) Let A be the set of pairs (α, x) such that Turing Machine Mα halts in at most
|x|3 steps on input x. Is A undecidable? Give a yes-or-no answer and justify it in one
sentence.
(b) (3 points) Define the term oblivious Turing Machine.
(c) (15 points) Construct an oracle O such that NPO is not equal to coNPO.
Question 2
For purposes of this question, use the fact that the IND-SET (independent set) problem is NP-
complete. Prove that the following three problems are also NP-complete.
(a) (5 points) VC (vertex cover): the set of all pairs (G, k), where G = (V, E) is a graph and
there is a subset V’ of V such that |V’| ≤ k and, for every edge (u, v) in E, at least one of u
or v is in V’.
(b) (5 points) SUBGRAPH-ISO (subgraph isomorphism): the set of all pairs (G, H) such that
G contains a subgraph that is isomorphic to H. An isomorphism f from graph G1 to graph
G2 is a bijective mapping from V(G1) to V(G2) such that, for all v1 and v2 in V(G1),
(v1, v2) ε E(G1) if and only if (f(v1), f(v2)) ε E(G2).
(c) (10 points) DOMINATING SET: the set of all pairs (G, k) such that there is a subset V’
of V(G) such that |V’| ≤ k and every vertex v in V(G) – V’ is adjacent to at least one
member of V’.
Question 3
(a) (7 points) Define implicitly logspace-computable reductions. Explain why they are
needed in the study of nondeterministic logspace.
(b) (7 points) Define read-once certificates. Explain why they are needed in the study of
nondeterministic logspace.
(c) (6 points) Prove that, for every space-constructible s(n) ≥ log n, NSPACE(s(n)) and
coNSPACE(s(n)) are contained in DTIME(2O(s(n))).
