CS 360 Introduction to the Theory of Computing
Spring 2008
Assignment 2 Solutions
1. Prove that the following languages are not regular:
1.
A
=
{
x
∈ {
0
,
1
}
*
:
the length of
x
is odd, and its middle symbol is 1
}
Solution:
We will use the pumping lemma to show
A
is nonregular. Assume
A
is regular.
Then there exists a pumping length
n
≥
1
such that every string
w
∈
A
, with

w
 ≥
n
, can
be written as
w
=
xyz
, where

xy
 ≤
n
,

y

>
0
, and
xy
i
z
∈
A
for all
i
≥
0
.
Let
w
= 0
n
10
n
,

w

= 2
n
+ 1
> n
.
Since
w
has odd length, and its middle symbol is
1, it’s in
A
. Then consider the decomposition of
w
as described in the pumping lemma:
y
is a nonempty susbstring of the first
n
characters of
w
. Since
w
starts with
n
zeroes, we
are guaranteed that
y
= 0
k
for some
1
≤
k
≤
n
. Then
xy
0
z
=
xz
= 0
n

k
10
n
must be in
A
.
However,
n > n

k
(since
k >
0
), and therefore, the middle symbol of the string is
not 1 (or, if
k
= 1
, then the string is not of the odd length), so
x /
∈
A
.
Contradiction.
A
is not regular.
2.
B
=
{
0
n
1
m
:
n, m
≥
0
are integers with
n
6
=
m
}
Solution 1:
We will use the pumping lemma to show
B
is nonregular. Assume
B
is regular.
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 Spring '08
 JohnWatrous
 Formal language, Formal languages, Regular expression, Regular language, ﬁrst n characters

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