COT 5310
Fall 2007
Midterm#2
Name:_______KEY___
12
1
.
Choosing from among
(REC)
recursive
,
(RE)
re nonrecursive
,
(CO) core nonrecursive
,
(NR)
nonre
, categorize each of the sets in a) through d). Justify your answer by showing some minimal
quantification of some known recursive predicate.
a.) { <f,x>  f(x) takes at least x
2
steps to converge }
REC
Justification: ~STP(x,f,x
2
1)
b.) { f  range(f) contains only even numbers }
CO
Justification:
∀
<x,t>[STP(x,f,t)
⇒
isEven(x)]
c.) { f

range(f) is not the set of natural numbers }
NR
Justification:
∃
x
∀
<y,t>[STP(y,f,t)
⇒
Value(y,f,t)
≠
x]
d.) { f  f converges on some pair of input, x, 2x
}
RE
Justification:
∃
<x,t> [STP(x,f,t) && STP(2x,f,t)]
9
2
. Let
A
be re, possibly recursive, and
B
be re nonrecursive. Let
C = (A
∩
~B)
∪
(B
∩
~A).
For each part, either show sets
A
and
B
with the specified property and
justify in detail
how these
meet the required property, or present a
demonstration
that this property cannot hold.
a.)
Can
C
be re nonrecursive?
YES. Let A =
φ
. A is clearly re, even recursive since it is trivially decided by
χ
A
(x) = 0. Then
C = (
φ
∩
~B)
∪
(B
∩
ℵ
)
=
B. B is given to be re, nonrecursive.
b.)
Can
C
be core nonrecursive?
YES. Let A =
ℵ
. A is clearly re, even recursive since it is trivially decided by
χ
A
(x) = 1.
Then C = (
ℵ
∩
~B)
∪
(B
∩
φ
)
= ~
B. Since B is given to be re, nonrecursive, its complement
must be core nonrecursive, as desired.
12
3
. Let set
A
and
B
be sets, such that
A
≤
m
B
by the total m1 recursive function
f
AB
.
For each of the
following, be
complete
by addressing whether or not the specified set can be recursive, re non
recursive and/or nonre.
a.)
Assume
A
is re, nonrecursive and semidecided by the partial recursive functions
g
A
. What can we
say about the complexity (recursive, re, nonre) of
B
? Address all three cases.
B is definitely not recursive and may not even be re.