COT 5310
Fall 2006
More Final Exam Samples
Name:
KEY
12
1
.
,
(?)
unknown
, categorize each of the
following decision problems. No proofs are required.
Problem / Language Class
Regular
Context Free
Context Sensitive
Choosing from among
(D)
decidable
,
(U)
undecidable
L =
Σ
* ?
D
U
U
L =
φ
D
D
U
?
L = L
2
?
D
U
U
x
∈
L
2
, for arbitrary x ?
D
D
D
8
2
.
,
(?)
unknown
, categorize each of the following closure
properties. No proofs are required.
Problem / Language Class
Regular
Context Free
Choosing from among
(Y)
yes
,
(N)
No
Closed under intersection?
Y
N
Closed under quotient?
Y
N
Closed under quotient with Regular languages?
Y
Y
Closed under complement?
Y
N
Consider the two operations on languages
max
and
min
max(L) = { x  x
∈
L
and, for no
y
does
xy
∈
L }
and
min(L) = { x  x
∈
L
and, for no prefix of
x
,
y
, does
y
∈
L }
D
L
1
m
8
3.
, where
escribe the languages produced by
max
and
min
. for each of the following:
=
{ a
i
b
j
c
k
 k
≤
i
or
k
≤
j }
ax(L
1
) =
{ a
i
b
j
c
k
 k =max(i, j)
}
in(L ) =
{
λ
} (string of length 0)
m
1
=
{ a
i
b
j
c
k
 k > i
or
k > j
}
ax(L
2
) =
{
} (empty)
L
2
m
min(L
2
) =
{ a
i
b
j
c
k
 k =min(i, j)+1
}
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View Full DocumentCOT 5310 Formal Languages and Automata Theory
– 2 –
Key for Final Samples – Hughes
12
4
.
Prove that any class of languages,
C
, closed under union, concatenation, intersection with regular
languages, homomorphism and substitution (e.g., the ContextFree Languages) is closed under
MissingMiddle
, where, assuming L is over the alphabet
Σ
,
MissingMiddle(L) = { xz 
∃
y
∈
Σ
* such that xyz
∈
L }
You must be very explicit, describing what is produced by each transformation you apply.
Define the alphabet
Σ
’ = { a’  a
∈Σ
}, where, of course, a’ is a “new” symbol, i.e., one not in
.
Define homomorphisms g and h, and substitution f as follows:
g(a) = a’
∀
a
h(a) = a
;
h(a’) =
λ
a
f(a) = {a, a’ }
a
Consider R =
*
•
g(
*)
* = { x y’ z  x, y, z
*
and
y’=g(y)
’* }
* is regular since it is the Kleene star closure of a finite set.
g(
*) is regular since it is the homomorphic image of a regular language.
R is regular as it is the concatenation of regular languages.
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