ASSIGNMENT #3
Due Wednesday, September 24, 2014
1. Consider the deterministic ﬁnite automaton M = (
{
q
1
,
q
2
,
q
3
}
,
{
0, 1
}
,
δ
,
q
1
,
{
q
2
}
) where
δ
is
deﬁned as follows:
δ
(
q
1
, 0) =
q
1
δ
(
q
1
, 1) =
q
2
δ
(
q
2
, 0) =
q
3
δ
(
q
2
, 1) =
q
2
δ
(
q
3
, 0) =
q
2
δ
(
q
3
, 1) =
q
2
Write an equivalent regular expression.
2. Prove that the following languages are not regular sets:
(a) L =
{
a
i
b
j
c
k

i
= 0
∨
j
=
k
,
i,j,k
≥
0
}
. Example strings include bccc, abbcc, aaa, etc.
(b) L =
{
ww

w
∈ {
0
,
1
}
+
}
. Example strings include 00, 11, 0101, 010010, etc.
(c) L =
{
a
2
n

n
≥
0
}
. Example strings include aaaa, a
16
, a
64
, etc.
(d) L =
{
w

w
∈ {
0
,
1
}
*
, w is of the form (0
i
1)
n
, for
i
= 1, 2, .
..,
n
,
n
≥
0
}
. The strings
of this language are
ε
, 01, 01001, 010010001, .
..., each successive string of 0’s being one
larger than the previous.
3. Find the minimum state ﬁnite automaton for the language speciﬁed by the ﬁnite automaton
M = (
{
q
0
,
q
1
,
q
2
,
q
3
}
,
{
0, 1
}
,
δ
,
q
0
,
{
q
0
}
) where
δ
is deﬁned as follows:
δ
(
q
0
, 0) =
q
3
δ
(
q
0
, 1) =
q
0
δ
(
q
1
, 0) =
q
0
δ
(
q
1
, 1) =
q
3
δ
(
q
2
, 0) =
q
2
δ
(
q
2
, 1) =
q
1
δ
(
q
3
, 0) =
q
1
δ
(
q
3
, 1) =
q
2