11711: Algorithms for NLP
Homework Assignment #1: Formal Language Theory
Solutions
Out: September 10, 2009
Due: September 24, 2009
Problem 1 [10 points]
Prove that, for any
deterministic
FSA
A
= (
Q,
Σ
, δ, q
0
, F
),
ˆ
δ
(
q, xy
) =
ˆ
δ
ˆ
δ
(
q, x
)
, y
for
x, y
∈
Σ
*
. Use the definition of
ˆ
δ
provided in lecture:
(1)
ˆ
δ
(
q,
) =
q
(2)
ˆ
δ
(
q, xσ
) =
δ
ˆ
δ
(
q, x
)
, σ
where
is the empty string,
x
∈
Σ
*
, and
σ
∈
Σ.
Solution
The proof is by induction on

y

.
Base:

y

= 0.
If

y

= 0, then
y
=
.
ˆ
δ
(
q, xy
)
=
ˆ
δ
(
q, x
)
by definition of
y
=
ˆ
δ
ˆ
δ
(
q, x
)
,
by definition (1) of
ˆ
δ
=
ˆ
δ
ˆ
δ
(
q, x
)
, y
by definition of
y
Induction:

y

=
n
+ 1
.
We rewrite
y
as
wσ
where
w
∈
Σ
*
and
σ
∈
Σ. Thus

w

=
n
, and we
assume by the inductive hypothesis that
ˆ
δ
(
q, xw
) =
ˆ
δ
ˆ
δ
(
q, x
)
, w
.
ˆ
δ
(
q, xy
)
=
ˆ
δ
(
q, xwσ
)
by definition of
y
=
δ
ˆ
δ
(
q, xw
)
, σ
by definition (2) of
ˆ
δ
=
δ
ˆ
δ
ˆ
δ
(
q, x
)
, w
, σ
by inductive hypothesis
=
ˆ
δ
ˆ
δ
(
q, x
)
, wσ
by definition (2) of
ˆ
δ
=
ˆ
δ
ˆ
δ
(
q, x
)
, y
by definition of
y
1