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Lecture 3: Subset construction and
±
NFAs
David Dill
Department of Computer Science
1
Outline
•
Proofs of equivalence of DFAs and NFAs
•
±
NFAs
•
Equivalence of
±
NFAs and DFAs.
2
Subset Construction
Given
N
= (
Q,
Σ
, q
0
, δ
N
, F
N
)
construct
D
= (2
Q
,
Σ
,
{
q
0
}
, δ
D
, F
D
)
such that
L
(
D
) =
L
(
N
)
.
The key is in the definition of
δ
D
and
F
D
:
where
δ
D
(
S, a
) =
S
q
∈
S
δ
N
(
q, a
)
, for each
S
∈
2
Q
.
The final states of the
D
are the subsets that contain at least one final state of
N
F
D
=
{
S
∈
2
Q

S
∩
F
N
6
=
∅}
3
Proof of the subset construction
Here is what we want:
Theorem
For every NFA
N
, if
D
is the DFA obtained by the subset construction,
L
(
N
) =
L
(
D
)
.
But it’s actually easier to prove a stronger claim:
Lemma
For every NFA
N
, if
D
is the DFA obtained by the subset construction,
ˆ
δ
D
(
{
q
0
}
, w
) =
ˆ
δ
N
(
q
0
, w
)
for all
w
∈
Σ
*
(I.e., the
state
you get to by running
D
on
w
is the same as the
set of states
you
get to by running
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This note was uploaded on 03/08/2011 for the course CS 154 taught by Professor Motwani,r during the Winter '08 term at Stanford.
 Winter '08
 Motwani,R
 Computer Science

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