University of Central Florida
School of Computer Science
COT 4210
Spring 2004
Prof. Rene Peralta
Homework 7: answers
1. The following is a hierarchy of subsets of
IN
.
regular
j
CFL
j
recursive
j
re.
j
P
(
IN
)
Show that the hierarchy is proper by giving examples of sets contained
in each level but not in the next one.
answer:
The question is poorly worded, apologies. But what I wanted
was explained in class:
1
• {
(2
n

1)2
n

1

n
≥
1
}
is contextfree but not regular.
• {
2
n
2

n
≥
0
}
is recursive but not contextfree.
•
Let
C
i
be the
i
th
C
program in some canonical ordering. Then
{
2
i
3
j

C
i
halts on input
j
}
is recursively enumerable but not re
cursive; and
{
2
i
3
j

C
i
loops on input
j
}
is not recursively enu
merable.
2. In this question, let
M
be a Turing machine which takes as input
a positive integer
i
and outputs (if it halts) a positive integer
M
(
i
).
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 Spring '08
 Staff
 Computer Science, Halting problem, Recursively enumerable language

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