Converting an NFA to a DFA

1

Lexical Analysis - Part 3

© Harry H. Porter, 2005

Converting an NFA to a DFAGiven:

A non-deterministic finite state machine (NFA)

Goal:

Convert to an equivalent deterministic finite state machine (DFA)

Why?

Faster recognizer!

Approach:

Consider simulating a NFA.

Work with sets of states.

IDEA: Each state in the DFA will correspond to a set of NFA states.

Worst-case:

There can be an exponential number O(2N) of sets of states.

The DFA can have exponentially many more states than the NFA

... but this is rare.

2



NFA to DFA

Input: A NFA S = States = { s0, s1, ..., sN} = SNFA

! = Move function = MoveNFA

Move’(S, a) " Set of states

Output: A DFA S = States = {?, ?, ..., ?} = SDFA

! = Move function = MoveDFA

Move(s, a) " Single state from SDFA

Main Idea: Each state in SDFA will be a set of states from the NFA

SDFA = { {...}, {...} , ..., {...} }

NFA DFA

(The names of the states is arbitrary and can be changed later, if desired.)

37

5

a

a

a{3} {5,7}

a1 2

3



Algorithm: Convert NFA to DFAWe’ll use...

MoveNFA(S,a) the transition function from NFA

#-Closure(s) where s is a single state from NFA

#-Closure(S) where S is a set of states from NFA

We’ll construct...

SDFA the set of states in the DFA

Initially, we’ll set SDFA to {}

Add X to SDFA where X is some set of NFA states

Example: “Add to SDFA”

We’ll “mark” some of the states in the DFA.

Marked = “We’ve done this one” ($)

Unmarked = “Still need to do this one”

MoveDFA(T,b) The transition function from DFA

To add an edge to the growing DFA...

Set MoveDFA(T,b) to S

...where S and T are sets of NFA states

{3,5,7}

bT S

4



Example

Start state:

#-Closure (0)

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

5



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7}

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

6



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

7



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)

=

MoveDFA(A,b)

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

8



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)

= #-Closure (MoveNFA(A,a))

=

MoveDFA(A,b)

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

9



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

=

MoveDFA(A,b)

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

10



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8}

MoveDFA(A,b)

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

11



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(A,b)

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

12



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(A,b)

= #-Closure (MoveNFA(A,b))

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

13



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(A,b)


= #-Closure ({5})

=

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

14



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(A,b)


= #-Closure ({5})

= {1,2,4,5,6,7} = C

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

15



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(A,b)


= #-Closure ({5})

= {1,2,4,5,6,7} = C

C

B

So far:

A a

b

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

16



Example

Start state:

#-Closure (0)

= {0, 1, 2, 4, 7} = A

MoveDFA(A,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(A,b)


= #-Closure ({5})

= {1,2,4,5,6,7} = C

A is now done; mark it!B and C are unmarked.Let’s do B next...

C

B

$A a

b

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

So far:

17



Process B = {1,2,3,4,6,7,8}

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

18



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)

=

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

19



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)

= #-Closure (MoveNFA(B,a))

=

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

20



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

=

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

21



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

22



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

a

23



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(B,b)

=

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

a

24



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(B,b)

= #-Closure (MoveNFA(B,b))

=

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

a

25



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(B,b)


= #-Closure ({5,9})

=

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

a

26



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(B,b)


= #-Closure ({5,9})

= {1,2,4,5,6,7,9} = D

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

C

B

$A a

b

a

27



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(B,b)


= #-Closure ({5,9})

= {1,2,4,5,6,7,9} = D

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

C

B

%= {a,b}

Db

a

$A

b

a

10

28



Process B = {1,2,3,4,6,7,8}

MoveDFA(B,a)


= #-Closure ({3,8})

= {1,2,3,4,6,7,8} = B

MoveDFA(B,b)


= #-Closure ({5,9})

= {1,2,4,5,6,7,9} = D

Example2

a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

C

B

%= {a,b}

D$

b

a

$A

b

a

10

29



Example

Process C = {1,2,4,5,6,7}

C

B

a

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

Db

30



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) =

MoveDFA(C,b) =

C

B

a

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

Db

31



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) =

C

B

a

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

Db

32



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

C

B

a

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

Db

33



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

C

B

a

a

b

$

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

Db

34



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) =

MoveDFA(D,b) =

C

B Db

a

a

b

$

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

35



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) =

C

B Db

a

a

b

$

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

36



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

a

b

$

$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

37



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

a

b

b

a

$

$$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

E

38



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

Process E = {1,2,4,5,6,7,10}

MoveDFA(E,a) =

MoveDFA(E,b) =

a

b

b

a

$

$$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

E

39



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

Process E = {1,2,4,5,6,7,10}

MoveDFA(E,a) = {1,2,3,4,6,7,8} = B

MoveDFA(E,b) =

a

b

b

a

$

$$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

E

40



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

Process E = {1,2,4,5,6,7,10}

MoveDFA(E,a) = {1,2,3,4,6,7,8} = B

MoveDFA(E,b) = {1,2,4,5,6,7} = C

a

b

b

a

$

$$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

E

41



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

Process E = {1,2,4,5,6,7,10}

MoveDFA(E,a) = {1,2,3,4,6,7,8} = B

MoveDFA(E,b) = {1,2,4,5,6,7} = C

a

b

b

a

a

b$

$ $$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

E

42



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

Process E = {1,2,4,5,6,7,10}

MoveDFA(E,a) = {1,2,3,4,6,7,8} = B

MoveDFA(E,b) = {1,2,4,5,6,7} = C

a

b

b

a

a

b$

$ $$$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

E

Final States in DFA?

...which state(s) contain 10?

43



Example

Process C = {1,2,4,5,6,7}

MoveDFA(C,a) = {1,2,3,4,6,7,8} = B

MoveDFA(C,b) = {1,2,4,5,6,7} = C

Process D = {1,2,4,5,6,7,9}

MoveDFA(D,a) = {1,2,3,4,6,7,8} = B

MoveDFA(D,b) = {1,2,4,5,6,7,10} = E

C

B Db

a

Process E = {1,2,4,5,6,7,10}

MoveDFA(E,a) = {1,2,3,4,6,7,8} = B

MoveDFA(E,b) = {1,2,4,5,6,7} = C

E

a

b

b

a

a

b$

$ $$

Final Result:

$A

b

a

2a

4b

15

3 #

##

#

60 # #

#

#

7 a8 b 9 b

%= {a,b}

10

Final States in DFA?

...which state(s) contain 10?

44



Algorithm: Convert NFA to DFASDFA = {}

Add #-Closure(s0) to SDFA as the start stateSet the only state in SDFA to “unmarked”

while SDFA contains an unmarked state do

Let T be that unmarked state

Mark T

for each a in % do S = #-Closure(MoveNFA(T,a)) if S is not in SDFA already then

Add S to SDFA (as an “unmarked” state)

endIf

Set MoveDFA(T,a) to S

endFor

endWhile

for each S in SDFA do

if any s&S is a final state in the NFA then Mark S an a final state in the DFA

endIf

endFor

Everywhere you could

possibly get to on an a

A set of NFA states

a{t1,t2,...} {s1,s2,...}

i.e, add an edge to the DFA...

T S

45



Resulting DFA for (a|b)*abb

Is it minimal?

C

BA a

b

Db

a

E

a

b

b

a

a

b

46




Is it minimal?

C

BA a

b

Db

a

E

a

b

b

a

a

b

BA a Db Eb

47




Is it minimal?

C

BA a

b

Db

a

E

a

b

b

a

a

b

BA a Db Eb

48




Is it minimal?

C

BA a

b

Db

a

E

a

b

b

a

a

b

BA a Db Eb

b

b

49




Is it minimal?

C

BA a

b

Db

a

E

a

b

b

a

a

b

BA a Db

a

Eb

a

a

b

b

50




Is it minimal?

C

BA a

b

Db

a

E

a

b

b

a

a

b

BA a Db

a

Eb

a

a

b

b

Every Regular Set is recognized

by a minimal DFA!

51




Is it minimal?

C

BA a

b

Db

a

E

a

b

b

a

a

b

BA a Db

a

Eb

a

a

b

b

Every Regular Set is recognized

by a minimal DFA!

And it is unique, up to renaming of states

Documents

Converting an NFA to a DFA