CS 181 — Winter 2005
Formal Languages and Automata Theory
Problem Set #4 Solutions
Problem 4.1.
(10 points)
1. Show that if all productions of a CFG
G
are of the form
A
→
wB
or
A
→
w
for variables
A,B
∈
V
and strings of terminals
w
∈
Σ
*
, then
L
(
G
) is regular.
Solution
Given a contextfree grammar
G
= (
V,
Σ
,R,S
) where each production
R
has either form
A
→
wB
or
A
→
w
, we will construct a GNFA for
L
(
G
). Here is the intuition: Our GNFA
will have a state for each variable in
V
an initial state
q
0
and a ﬁnal state
q
f
. Our initial
state
q
0
has a transition into
S
on
ε
. For each production of the form
A
→
wB
we have a
transition from the state
A
to the state
B
over regular expression
w
. For each production of
the form
A
→
w
we have a transition from the state
A
to the state
q
f
over
w
.
Formally, our GNFA is deﬁned as
M
= (
V
∪ {
q
0
,q
f
}
,
Σ
,δ,q
0
,
{
q
f
}
) where the transition
function is
δ
(
q,p
) =
ε,
if
q
=
q
0
and
p
=
S
S
w
:(
q
→
w
)
∈
R
w,
if
q
∈
V
and
p
=
q
f
S
w
:(
q
→
wp
)
∈
R
w,
if
q,p
∈
V
∅
,
otherwise
Notice that to handle cases where our grammar has productions
A
→
w
1
B
and
A
→
w
2
B
,
we have a transition from
A
to
B
over
w
1
∪
w
2
.
Let us now show that in fact
L
(
M
) =
L
(
G
). Suppose
w
∈
L
(
G
) then there is some derivation
S
=
⇒
w
1
V
1
=
⇒
w
1
w
2
V
2
=
⇒ ···
=
⇒
w
1
w
2
···
w
k

1
V
k

1
=
⇒
w
1
w
2
···
w
k
=
w
Then clearly the run of
M
on
w
with state sequence
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 Spring '08
 griebach
 Formal language, Formal languages, Regular expression, Contextfree grammar, contextfree language, R R R R

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