CS 341 Automata Theory
Elaine Rich
Homework 10
Due Thursday, Nov. 9 at 11:00
1)
Determine, for each of the following languages, whether it is (I) Regular, (II) Context free but not regular, or
(III) not context free.
Prove your answer.
a)
L
= {
ww
R
w
:
w
∈
{
a
,
b
}*}.
b)
L
= {
w
:
w
=
uu
R
or
w
=
u
a
n
:
n
= 
u
,
u
∈
{
a
,
b
}*}.
c)
L
= {
a
n
b
2
n
c
m
}
∩
{
a
n
b
m
c
2
m
}.
d)
L
*, where
L
= {
0
*
1
i
0
*
1
i
0
*
}.
Hint: This one is tricky.
Be very careful in your analysis of
L
*.
e)
L
= {
w
∈
{0, 1}* : #
1
(
w
) = (#
0
(
w
))
2
}
f)
L
= {
x
∈
{
a
,
b
}* : 
x
 is even and the first half of
x
has one more
a
than does the second half}
2)
Let
L
1
= {
a
n
b
m
:
n
≥
m
}.
Let
R
1
= {(
a
∪
b
)* : there is an odd number of
a
's and an even number of
b
's}.
Use
the construction described in the book to build a PDA that accepts
L
1
∩
R
1
.
3)
Suppose that
L
is context free and
R
is regular.
Is
R

L
necessarily context free?
Prove your answer.
4)
Show that the contextfree languages are closed under letter substitution.
5)
Let
alt
be a function that maps from any language
L
over some alphabet
Σ
to a new language
L
′
as follows:
alt
(
L
) = {
x
:
5
y,n
(
y
∈
L
, 
y
 =
n
,
n
> 0,
y
=
a
1
…
a
n
,
2200
i
≤
n
(
a
i
∈
Σ
), and
x
=
a
1
a
3
a
5
…
a
k
, where
k
= (if even
(
n
) then
n
1 else
n
))}
a)
Consider
L
=
a
n
b
n
.
Clearly describe
L
1
=
alt
(
L
).
b)
Are the context free languages closed under the function
alt
?
Prove your answer.
6)
Let
chop
be a function that maps from any language
L
over some alphabet
Σ
to a new language
L
′
as follows:
chop
(
L
) = {
x
:
5
y
∈
L
(
y
=
uvw
,
u
,
w
∈
Σ
*,
v
∈
Σ
, 
u
 = 
w
, and
x
=
uw
)}
In other words, the strings in
chop
(
L
) are the odd length strings in
L
with the middle character chopped out.
Prove that the context free languages are not closed under
chop
.
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View Full DocumentCS 341 Automata Theory
Elaine Rich
Homework 10
Answers
1)
Determine, for each of the following languages, whether it is (I) Regular, (II) Context free but not regular, or
(III) not context free.
Prove your answer.
a)
L
= {
ww
R
w
:
w
∈
{
a
,
b
}*}
(III) Not Context Free.
Use pumping:
Let
w
=
a
k
b
k
b
k
a
k
a
k
b
k
1

2

3

4
In each of these cases, pump in once:
1.
If any part of
v
is in region 1, then to produce a string in
L
we must also pump
a
’s into region 3.
But
we cannot since 
vxy

≤
k
.
2.
If any part of
v
is in region 2, then to produce a string in
L
we must also pump
b
’s into region 4.
But
we cannot since 
vxy

≤
k
.
3.
If any part of
v
is in region 3, then to produce a string in
L
we must also pump
a
’s into region 1.
But
we cannot since
y
must come after
v
.
4.
If any part of
v
is in region 4, then to produce a string in
L
we must also pump
b
’s into region 2.
But
we cannot since y must come after
v
.
b)
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 Fall '08
 Rich
 Pump, Formal language, CF, Half, context free languages

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