CS 341 Automata Theory
Elaine Rich
Homework 5
Due: Thursday, February 15, 2007
This assignment covers Chapters 8 and 9.
1)
For each of the following languages
L
, state whether or not
L
is regular.
Prove your answer:
a)
{
a
i
b
j
:
i, j
≥ 0 and
i
+
j
= 5}.
Regular.
A simple FSM with five states just counts the total number of characters.
b)
{
a
i
b
j
:
i, j
≥ 0 and
i

j
= 5}.
Not regular.
L
consists of all strings of the form
a
*
b
* where the number of
a
’s is five more than the
number of
b
’s.
We can show that
L
is not regular by pumping.
Let
w
=
a
k
+5
b
k
.
Since 
xy

≤
k
,
y
must equal
a
p
for some
p
> 0.
We can pump
y
out once, which will generate the string
a
k
+5
p
b
k
, which is not in
L
because the number of
a
’s is is less than 5 more than the number of
b
’s.
c)
{
w
=
xy
:
x
,
y
∈
{
a
,
b
}* and 
x
 = 
y
 and #
a
(
x
)
≥
#
a
(
y
)}.
Not regular, which we’ll show by pumping.
Let
w
=
a
k
bba
k
.
y
must occur in the first
a
region and be
equal to
a
p
for some nonzero
p
.
Let
q
= 0.
If
p
is odd, then the resulting string is not in
L
because all
strings in
L
have even length.
If
p
is even it is at least 2.
So both
b
’s are now in the first half of the string.
That means that the number of
a
’s in the second half is greater than the number in the first half.
So
resulting string,
a
kp
bba
k
, is not in
L
.
d)
{
w
=
xyzy
R
x
:
x
,
y
,
z
∈
{
a
,
b
}*}.
Regular.
Note that
L
= (
a
∪
b
)*.
Why?
Take any string
s
in (
a
∪
b
)*.
Let
x
and
y
be
ε
.
Then
s
=
z
.
So
the string can be written in the required form.
Moral: Don’t jump too fast when you see the nonregular
“triggers”, like
ww
or
ww
R
.
The entire context matters.
e)
{
w
=
xyzy
:
x
,
y
,
z
∈
{
0
,
1
}
+
}.
Regular.
All that’s required is that the last letter must occur in at least one other place in the string, that
place is not immediately to its left, and thereare at least 4 letters.
Either the repeated character is
0
or
1
.
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 Fall '08
 Rich
 Formal language, Regular expression, L′

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