Nondeterministic Finite Automata
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Transcript Nondeterministic Finite Automata
Non-Deterministic
Finite Automata
Fall 2006
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Nondeterministic Finite Automaton (NFA)
Alphabet = {a}
a
q0
q1 a
q2
a
q3
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Alphabet = {a}
Two choices
a
q0
q1 a
q2
a
q3
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Alphabet = {a}
Two choices
a
q0
q1 a
No transition
a
q3
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q2
No transition
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First Choice
a a
a
q0
q1 a
q2
a
q3
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First Choice
a a
a
q0
q1 a
q2
a
q3
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First Choice
a a
All input is consumed
a
q0
q1 a
q2
“accept”
a
q3
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Second Choice
a a
a
q0
q1 a
q2
a
q3
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Second Choice
a a
Input cannot be consumed
a
q0
a
q1 a
Automaton Halts
q3
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q2
“reject”
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An NFA accepts a string:
if there is a computation of the NFA
that accepts the string
i.e., all the input string is processed and the
automaton is in an accepting state
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aa is accepted by the NFA:
“accept”
a
q0
q1 a
q2
a
q0
a
q3
because this
computation
accepts aa
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q1 a
a
q3
q2
“reject”
this computation
is ignored
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Rejection example
a
a
q0
q1 a
q2
a
q3
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First Choice
a
“reject”
a
q0
q1 a
q2
a
q3
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Second Choice
a
a
q0
q1 a
q2
a
q3
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Second Choice
a
a
q0
q1 a
a
q3
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q2
“reject”
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Another Rejection example
a a a
a
q0
q1 a
q2
a
q3
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First Choice
a a a
a
q0
q1 a
q2
a
q3
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First Choice
a a a
Input cannot be consumed
a
q0
q1 a
a
q3
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q2
“reject”
Automaton halts
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Second Choice
a a a
a
q0
q1 a
q2
a
q3
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Second Choice
a a a
Input cannot be consumed
a
q0
q1 a
Automaton halts
a
q3
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q2
“reject”
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An NFA rejects a string:
if there is no computation of the NFA
that accepts the string.
For each computation:
• All the input is consumed and the
automaton is in a non final state
OR
• The input cannot be consumed
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a
is rejected by the NFA:
“reject”
a
q0
q1 a
q2
a
q0
a
q3
“reject”
q1 a
q2
a
q3
All possible computations lead to rejection
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aaa
is rejected by the NFA:
“reject”
a
q0
q1 a
q2
a
q0
a
q3
q1 a
a
q3
q2
“reject”
All possible computations lead to rejection
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Language accepted:
a
q0
L {aa}
q1 a
q2
a
q3
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Lambda Transitions
q0 a
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q1
q2 a
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q3
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a a
q0 a
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q1
q2 a
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q3
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a a
q0 a
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q1
q2 a
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q3
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input tape head does not move
a a
q0 a
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q1
q2 a
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q3
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all input is consumed
a a
“accept”
q0 a
String
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q1
q2 a
q3
aa is accepted
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Rejection Example
a a a
q0 a
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q1
q2 a
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q3
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a a a
q0 a
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q1
q2 a
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q3
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(read head doesn’t move)
a a a
q0 a
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q1
q2 a
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q3
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Input cannot be consumed
a a a
Automaton halts
“reject”
q0 a
String
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aaa
q1
q2 a
q3
is rejected
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Language accepted:
q0 a
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q1
L {aa}
q2 a
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q3
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Another NFA Example
q0
a
b
q1
q2
q3
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a b
q0
a
b
q1
q2
q3
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a b
q0
a
b
q1
q2
q3
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a b
“accept”
q0
a
b
q1
q2
q3
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Another String
a b a b
q0
a
b
q1
q2
q3
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a b a b
q0
a
b
q1
q2
q3
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a b a b
q0
a
b
q1
q2
q3
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a b a b
q0
a
b
q1
q2
q3
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a b a b
q0
a
b
q1
q2
q3
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a b a b
q0
a
b
q1
q2
q3
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a b a b
“accept”
q0
a
b
q1
q2
q3
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Language accepted
L ab, abab, ababab, ...
ab
q0
a
b
q1
q2
q3
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Another NFA Example
0
q0
1
q1
0, 1 q2
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Language accepted
L(M ) = {λ, 10, 1010, 101010, ...}
= {10} *
0
q0
1
q1
0, 1 q2
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(redundant
state)
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Remarks:
•The symbol never appears on the
input tape
•Simple automata:
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M1
q0
M2
L(M1) = {}
L(M 2 ) = {λ}
q0
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•NFAs are interesting because we can
express languages easier than DFAs
NFA
q0
a
DFA
M1
q2
q1
a
q0
L( M1) = {a}
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a
M2
a
q1
L( M 2 ) = {a}
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Formal Definition of NFAs
M Q, , , q0 , F
Q : Set of states, i.e. q0 , q1, q2
: Input aplhabet, i.e. a, b
:
q0 :
Transition function
Initial state
F : Accepting states
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Transition Function
q , x q1, q2,, qk
q
x
x
q1
q1
x
resulting states with
following one transition
with symbol x
qk
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q0 , 1 q1
0
q0
1
q1
0, 1 q
2
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(q1,0) {q0 , q2}
0
q0
1
q1
0, 1 q
2
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(q0 , ) {q2 }
0
q0
1
q1
0, 1 q
2
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(q2 ,1)
0
q0
1
q1
0, 1 q
2
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Extended Transition Function
Same with
*
but applied on strings
q0, a q1
*
q5
q4
a
q0
a
a
b
q1
q2
q3
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q0, aa q4, q5
*
q5
q4
a
q0
a
a
b
q1
q2
q3
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q0, ab q2, q3, q0
*
q5
q4
a
q0
a
a
b
q1
q2
q3
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Special case:
for any state q
q q ,
*
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In general
q j qi ,w : there is a walk from qi to q j
with label w
*
w
qi
qj
w 1 2 k
qi
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1
k
2
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qj
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The Language of an NFA M
The language accepted by
M
is:
LM w1,w2,...wn
where
(q0 ,wm ) {qi ,...,qk ,, q j }
*
and there is some
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qk F
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(accepting state)
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wm LM
(q0,wm )
*
wm
qi
q0 w
m
qk
wm
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qk F
qj
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F q0 ,q5
q5
q4
a
a
q0
a
b
q1
q2
q3
q0 , aa q4 , q5
*
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aa L(M )
F
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F q0 ,q5
q5
q4
a
a
q0
a
b
q1
q2
q3
q0 , ab q2, q3, q0
*
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ab LM
F
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F q0 ,q5
q5
q4
a
q0
a
a
b
q1
q2
q3
q0 , abaa q4 , q5
*
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aaba L(M )
F
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F q0 ,q5
q5
q4
a
q0
a
a
b
q1
q2
q3
* q0, aba q1
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F
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aba LM
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q5
q4
a
q0
a
a
b
q1
q2
q3
LM ab * ab * {aa }
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NFAs accept the Regular
Languages
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Equivalence of Machines
Definition:
Machine
if
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M1
is equivalent to machine
M2
LM1 LM 2
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Example of equivalent machines
NFA
LM1 {10}*
0
q0
q1
1
DFA
LM 2 {10}*
M1
M2
0,1
0
q0
1
q1
1
q2
0
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Theorem:
Languages
accepted
by NFAs
Regular
Languages
Languages
accepted
by DFAs
NFAs and DFAs have the same computation power,
accept the same set of languages
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Proof:
we only need to show
Languages
accepted
by NFAs
Regular
Languages
AND
Languages
accepted
by NFAs
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Regular
Languages
73
Proof-Step 1
Languages
accepted
by NFAs
Regular
Languages
Every DFA is trivially an NFA
Any language L accepted by a DFA
is also accepted by an NFA
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Proof-Step 2
Languages
accepted
by NFAs
Regular
Languages
Any NFA can be converted to an
equivalent DFA
Any language L accepted by an NFA
is also accepted by a DFA
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Conversion NFA to DFA
a
NFA M
a
q
q
q
0
1
2
b
DFA
M
q0
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NFA
* (q0, a ) {q1, q2 }
a
M
q0
a
q1
q2
b
DFA
M
q0
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a
q1, q2
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NFA
* (q0 , b )
a
M
q0
a
empty set
q1
q2
b
DFA
M
q0
a
q1, q2
trap state
b
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(q1, a ) {q1, q2 }
* (q2, a )
*
NFA
a
M
q0
a
q1
union
q2
q1, q2
b
DFA
a
M
q0
a
q1, q2
b
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(q1, b ) {q0 }
* (q2, b ) {q0 }
*
NFA
a
M
q0
a
q1
union
q2
q0
b
DFA
M
a
b
q0
a
q1, q2
b
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NFA
a
M
q0
a
q1
q2
b
DFA
M
a
b
q0
a
q1, q2
b
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a, b trap state
81
END OF CONSTRUCTION
NFA
a
M
q0
a
q1
q1 F
q2
b
DFA
M
a
b
q0
a
q1, q2
b
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q1, q2 F
a, b
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General Conversion Procedure
Input: an NFA
M
Output: an equivalent DFA M
with LM L(M )
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The NFA has states
q0 , q1, q2 ,...
The DFA has states from the power set
, q0 , q1 , q0 , q1 , q1, q2, q3, ....
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Conversion Procedure Steps
step
1.
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Initial state of NFA:
q0
Initial state of DFA:
q0
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Example
NFA
a
M
q0
a
q1
q2
b
DFA
M
q0
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step
2. For every DFA’s state {qi , q j ,..., qm}
compute in the NFA
* qi , a
* q j , a
...
Union
{qk , ql,...,qn }
* qm , a
add transition to DFA
{qi , qj ,...,qm }, a {qk , ql,...,qn }
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Example
NFA
* (q0 , a) {q1, q2}
a
M
a
q0
q1
q2
b
DFA
M
q0 , a q1, q2
q0
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a
q1, q2
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step
3. Repeat Step 2 for every state in DFA and
symbols in alphabet until no more states
can be added in the DFA
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Example
NFA
a
M
q0
a
q1
q2
b
DFA
M
a
b
q0
a
q1, q2
b
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a, b
90
step
4. For any DFA state {qi , q j ,..., qm}
if some
q j is accepting state in NFA
Then, {qi , q j ,..., qm }
is accepting state in DFA
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Example
NFA
a
M
q0
a
q1
q1 F
q2
b
DFA
M
a
b
q0
a
q1, q2
b
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q1, q2 F
a, b
92
Lemma:
If we convert NFA M to DFA M
then the two automata are equivalent:
L M L M
Proof:
We only need to show:
LM LM
AND
LM LM
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First we show:
LM LM
We only need to prove:
w L(M )
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w L(M )
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NFA
Consider
w L(M )
w
q0
qf
symbols
w 1 2 k
q0
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1
2
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k
qf
95
symbol
qi
i
qj
denotes a possible sub-path like
qi
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symbol
i
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qj
96
We will show that if
w L(M )
w 1 2 k
NFA
M:
q0
1
k
2
qf
then
DFA
1
M:
{q0}
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state
label
2
w L(M )
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k
{q f ,}
state
label
97
More generally, we will show that if in
(arbitrary string)
NFA
M:
q0
M:
v a1a2 an
a1
qi
a2
qj
ql
an
qm
then
DFA
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M:
a1
a2
{q0} {qi ,} {q j ,}
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an
{ql ,} {qm ,}
98
Proof by induction on
v a1
Induction Basis: |v | 1
NFA
M:
DFA
M:
q0
|v|
a1
qi
a1
{q0}
{qi ,}
is true by construction of M
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Induction hypothesis:
1 | v | k
v a1a2 ak
Suppose that the following hold
NFA
M:
DFA
M:
q0
a1
{q0}
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a1
qi
a2
qj
a2
{qi ,} {q j ,}
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qc
ak
qd
ak
{qc ,} {qd ,}
100
| v | k 1
v a1a2 ak ak 1 vak 1
Induction Step:
v
Then this is true by construction of M
NFA
M:
q0
a1
qi
a2
qj
qc
ak
qd
ak 1
qe
v
DFA
M:
a1
{q0}
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ak
a2
{qi ,} {q j ,}
v
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ak 1
{qc ,} {qd ,}
{qe ,}
101
Therefore if
w L(M )
w 1 2 k
NFA
M:
1
q0
k
2
qf
then
DFA
M:
1
{q0}
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2
w L(M )
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k
{q f ,}
102
We have shown:
With a similar proof
we can show:
Therefore:
LM LM
LM LM
LM LM
END OF LEMMA PROOF
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