CS 535  Fall 2008  MIDTERM with Answers
DIRECTIONS: Do any 4 of the following 5 problems. Each problem is worth
10 points. Write all of your answers in your blue book. The test is open book
and you can use one page of notes.
1. a. Let f be a total computable function, f:
N
→
N
. For any integer
y
,
f
−
1
(
y
) is defined to be the (possibly infinite) set of all
x
′
s
such that
f
(
x
) =
y
.
Show that for any fixed integer
m
,
f
−
1
(
m
) is a c.e. set.
Answer. If
f
−
1
(
y
) is finite then it is c.e by definition.
If
f
−
1
(
y
) is infinite we define a computable function h which enumerates
f
−
1
(
y
) by,
h(0) = smallest number
z
0
with f(
z
0
) = m
For j
>
0, h(j) = smallest number z larger than h(j1) with f(
z
) =m.
h is a total computable function which enumerates
f
−
1
(
m
).
b.
Given two disjoint computably enumerable sets,
A
and
B
, prove
that if
A
is undecidable then
A
∪
B
is also undecidable.
Ans.
Assume
A
∪
B
is decidable. , We will get a contradiction to this as
sumption by showing that in this case that A is decidable.
Given any input x, to decide if
x
∈
A
, first check if
x
∈
A
∪
B
,
If no, then x
/
∈
A.
If yes, then enumerate A and B until x appears in one (and only one) of these
sets.
Then
x
∈
A
if and only if x appears in A’s enumeration.
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 Fall '09
 Graph Theory, Data Mining, Halting problem, Bipartite graph, Tape head, computable function, NDTM

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