COT 6410
Fall 2010
Final Exam Sample Questions
1
.
Let set
A
be recursive,
B
be re nonrecursive and
C
be nonre. Choosing from among
(REC)
recursive
,
(RE)
re nonrecursive
,
(NR)
nonre
, categorize the set
D
in each of a) through d) by
listing
all
possible categories. No justification is required.
a.)
D = ~C
b.) D
⊆
(A
∪
C)
c.)
D = ~B
d.) D = B

A
2.
Prove that the
Halting Problem
(the set
K
0
)
is not recursive (decidable) within any formal model
of computation. (Hint: A diagonalization proof is required here.)
3.
Using reduction from the known undecidable
HasZero, HZ = { f 
5
x f(x) = 0 }
, show the non
recursiveness (undecidability) of the problem to decide if an arbitrary primitive recursive function
g
has the property
IsZero, Z = { f 
2200
x f(x) = 0 }
.
4
.
Choosing from among
(D)
decidable
,
(U)
undecidable
,
(?)
unknown
, categorize each of the
following decision problems. No proofs are required.
Problem / Language Class
Regular
Context Free
Context Sensitive
L =
Σ
* ?
L =
φ
?
L = L
2
?
x
∈
L
2
, for arbitrary x ?
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 Fall '10
 Dutton
 Halting problem, decision problem, Recursion theory, Primitive recursive function

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