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CS 341 Automata Theory
Elaine Rich
Homework 4
Due: Thursday, February 8, 2007
1)
Consider the following FSM
M
:
a
a
b
1
b
2
a
3
b
4
a,b
a)
Show a regular expression for
L
(
M
).
(
a
∪
bb
*
aa
)* (
ε
∪
bb
*(
a
∪
ε
)).
b)
Describe
L
(
M
) in English.
All strings in {
a
,
b
}* that contain no occurrence of
bab
.
2)
Consider the following FSM
M
:
q
0
a
q
1
b
q
3
b
a
b
q
2
a)
Write a regular expression that describes
L
(
M
).
ε
∪
((
a
∪
ba
)(
ba
)*
b
).
b)
Show a DFSM that accepts
¬
L
(
M
).
a
{1}
{2, 3}
b
{0}
a
b
a
{2}
∅
a,b
b
3)
Show a regular grammar for each of the following languages:
a)
{
w
∈
{
a
,
b
}* :
w
does not end in
aa
}.
S
→
a
A

b
B

ε
A
→
a
C

b
B

ε
B
→
a
A

b
B

ε
C
→
a
C

b
B
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{
w
∈
{
a
,
b
}* :
w
does not contain the substring
aabb
}.
S
→
a
A
A
→
a
B
B
→
a
B
C
→
a
A
S
→
b
S
A
→
b
S
B
→
b
C
C
→
ε
S
→
ε
A
→
ε
B
→
ε
4)
Consider the following regular grammar
G
:
S
→
a
T
T
→
b
T
T
→
a
T
→
a
W
W
→
ε
W
→
a
T
a)
Write a regular expression that generates
L
(
G
).
b)
Use
grammartofsm
to generate an FSM
M
that accepts
L
(
G
).
5)
Prove
by construction
that the regular languages are closed under:
a)
intersection.
b)
set difference.
6)
Prove that the regular languages are closed under each of the following operations:
a)
pref
(
L
) = {
w
:
5
x
∈
Σ
* (
wx
∈
L
)}.
By construction.
Let
M
= (
K
,
Σ
,
δ
,
s
,
A
) be any FSM that accepts
L
:
Construct
M′
= (
K
′
,
Σ′
,
δ′
,
s
′
,
A
′
) to accept
pref
(
L
) from
M
:
1) Initially, let
M′
be
M
.
2) Determine the set
X
of states in
M′
from which there exists at least one path to some accepting state:
a) Let
n
be the number of states in
M′
.
b) Initialize
X
to {}.
c) For each state
q
in
M′
do:
For each string
w
in
Σ
where 0 ≤
w
≤
n
1 do:
Starting in state
q
, read the characters of
w
one at a time and make the
appropriate transition in
M′
.
If this process ever lands in an accepting state of
M′
, add
q
to
X
and quit
processing
w
.
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 Fall '08
 Rich

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