課程名稱︰資訊工程理論基礎
課程性質︰必修
課程教師：呂育道
開課學院：電資學院
開課系所︰資工所
考試日期（年月日）︰2011.11.08
考試時限（分鐘）：
試題 :
Theory of Computation
Mid-Term Examination on November 8, 2011
Fall Semester, 2011
Note: You may use any result proved in class.
Problem 1 (30 points)
It is known that 3-COLORING is NP-complete. Show that 6-COLORING is
NP-complete. (You do not need to show that it is in NP.)
Ans: We reduce 3-COLORING to 6-COLORING (the problem of asking if a graph can
be colored by 6 or fewer colors such that no adjacent nodes have the same
color). Given a graph G(V, E) for 3-COLORING, the reduction outputs a
graph G'(V', E') by adding 3 new nodes with edges between each of the 3
nodes and all the other nodes in V. That is, V' = V ∪ {x_1, x_2, x_3}
and E' = E ∪ {(x_i, v)│ v ∈ V', i = 1, 2, 3, x_i ≠ v}. If
G ∈ 3-COLORING, then G' ∈ 6-COLORING because 3 or fewer colors for the
nodes in V and additional 3 colors for those in {x_1, x_2, x_3} suffice
to make no adjacent nodes have the same color. Conversely, consider a
legal coloring of G' with 6 or fewer colors. In such a coloring,
{x_1, x_2, x_3} use up exactly 3 colors, leaving at most 3 colors for the
nodes in V.
Problem 2 (30 points)
Let A → B denote the set of functions from set A to set B.
(a) [15 points] How many functions in {0, 1, 2, 3}^n → {0, 1} are there?
(b) [15 points] How many functions in ({0, 1, 2, 3}^n → {0, 1}) → {0, 1, 2}^m
are there? (Do not write something like x^a^b as it is ambiguous. Write
x^(a^b) or (x^a)^b.)
Ans: (a) 2^(4^n).
(b) (3^m)^(2^(4^n)).
Problem 3 (15 points)
Show that if L and ﹊L (﹊L: 代表L上面加一條線) are recursively enumerable,
then L is recursive.
Ans: Suppose that L and ﹊L are accepted by the one-string Turing machines
M and ﹊M, respectively. Then L is decided by a two-string Turing machine
M', defined as follows. On input x, M' simulates, on two different
strings, M and ﹊M on x in an interleaved fashion. That is, it simulates
a step of M, then a step of ﹊M, then again another step of M, and so on.
Since x is a member of L or ﹊L (but not both), exactly one of the two
machines will halt and accept. If M accepts, then M' halts on state
"yes". If ﹊M accepts, then M' halts on "no".
Problem 4 (25 points)
Let L denote the language {M: M halts on all inputs}. Show that L is not a
recursive language, that is, membership in L is undecidable.
Ans: We know the Halting Problem H = {M;x: M(x) ≠ ↗} is undecidable. Given
the question "M;x ∈ H?", we construct the machine: M_x(y) : M(x). M_x
halts on all inputs if and only if M halts on x. In other words, M_x ∈ L
if and only if M;x ∈ H. So if L were recursive, H would be recursive, a
contradiction.