CS 181 — Winter 2005
Formal Languages and Automata Theory
Problem Set #2 Solutions
Problem 2.1.
Convert the following
NFA
into an equivalent
DFA
. (Don’t bother to write down
unreachable states.) Also, describe in words the language accepted by each automaton.
A
B
0,1
C
1
0,1
D
1
Solution
A
AB
1
0
0
ABC
1
1
0
0
1
1
0
ABCD
AD
To build this machine, we simply use the subset construction. Starting from the initial state
A
, on input
0
we can only stay in
A
. Thus, the state
{
A
}
goes to itself on
0
. From
A
, on
input
1
, we can go to either
A
or
B
. Thus, the state
{
A
}
goes to
{
A, B
}
on input
1
. We
continue this process
{
A, B
}
and all subsequent states. The final states are the subsets that
contain final states. Thus,
{
A, B, C, D
}
and
{
A, D
}
are final states of the DFA.
Looking at the NFA, we can see that this machine only looks at the last three symbols of
input. While the last doesn’t matter, the second and third from last are required to be
1
’s.
After making these obsevations, it is easy to see that the language of the NFA is the set of
strings whose second and third from last symbols are
1
’s.
We can also deduce this from the DFA by giving the states the following interpretations:
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{
A
}
. . .
string is
ε
or ends with 010 or 00
{
A, B
}
. . .
string is 1 or ends with 01
{
A, B, C
}
. . .
string is 11 or ends with 011
{
A, B, C, D
}
. . .
string ends with
111
{
A, D
}
. . .
string ends with
110
Problem 2.2.
Say that string
x
is a
prefix
of string
y
if a string
z
exists where
xz
=
y
and that
x
is a
proper prefix
of
y
if in addition
x
6
=
y
. In each of the following parts, we define an operator
on a language
A
. Show that the class of regular languages is closed under the operation.
1.
NOPREFIX
(
A
) =
{
w
∈
A

no proper prefix of
w
is a member of
A
}
Solution
Let
M
= (
Q,
Σ
, δ, q
0
, F
) be a DFA for
A
. The key observation is the following: Let
w
∈
A
and
say the run of
M
on
w
is given by state sequence
r
0
, r
1
, . . . , r
n
. Then if
w
∈
NOPREFIX
(
A
),
then
w
has no proper prefix that is also in
A
. In particular, this means that none of the states
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 Spring '08
 griebach
 Formal language, Regular expression, Regular language, final state

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