This FA
0
0
1
0
,
1
q
0
q
1
q
2
In this ﬁnite automaton:
•
Q
=
{
q
0
,
q
1
,
q
2
}
,
•
Σ
=
{
0
,
1
}
,
•
δ
is a function from
Q
×
Σ
→
Q
. It includes
(
q
0
,
0
)
→
q
1
•
q
0
=
q
0
•
F
=
{
q
3
}
.
This FA
accepts
the
language
of words with 00 as a substring.
2.5
Acceptance, extension
Given a FA
M
, what does “
M
accepts
w
” mean?
•
Starting at
q
0
, follow transition function
δ
for each letter in
w
, in turn.
•
String
w
accepted by
M
if at the end of
w
’s transitions, we wind up in a state from
F
.
More formal deﬁnition comes by looking at the extended transition function,
ˆ
δ
.
•
ˆ
δ
(
q
,
w
)
: state we wind up in if we start at state
q
and follow
δ
for each letter in
w
in turn.
2.6
Formal deﬁnition of extended transition function
Formally,
ˆ
δ
(
q
,
w
)
is deﬁned recursively:
•
ˆ
δ
(
q
,
ε
) =
q
, for all states
q
.
•
If

w

>
0, and
w
=
xa
, then
ˆ
δ
(
q
,
w
) =
δ
(
ˆ
δ
(
q
,
x
)
,
a
)
.
•
Here,
x
is all but the last letter of
w
, and
a
is the last letter of
w
.
Unpack that:
•
q
1
=
ˆ
δ
(
q
,
x
)
: where we get from
q
when we read
x
.
•
Then, process letter
a
:
δ
(
q
1
,
a
) =
δ
(
ˆ
δ
(
q
,
x
)
,
a
)
.
Can be deﬁned from the other end:
•
If
w
=
ax
, then
ˆ
δ
(
q
,
w
) =
ˆ
δ
(
δ
(
q
,
a
)
,
x
)
.
•
Textbook gives the ﬁrst deﬁnition.
2.7
Language of an FA
The
language
of the ﬁnite automaton
M
: all words
M
accepts.
Formally, acceptance of a word:
•
M
= (
Q
,
Σ
,
δ
,
q
0
,
F
)
accepts
w
exactly when
ˆ
δ
(
q
0
,
w
)
∈
F
.
Language of the FA: all such words.
•
L
(
M
) =
{
w
∈
Σ
*
where
ˆ
δ
(
q
0
,
w
)
∈
F
}
.
•
(or just
{
w
∈
Σ
*
where
M
accepts
w
}
)
Terminlogy:
L
(
M
)
can be called:
•
The language of the FA
•
The language
accepted
by the FA
•
The language
recognized
by the FA
2.8
2