Convert to an equivalent deterministic finite state machine (DFA). Why? ... IDEA:
Each state in the DFA will correspond to a set of NFA states. Worst-case:.
Lexical Analysis - Part 3
Converting an NFA to a DFA Given: 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. © Harry H. Porter, 2005
1
Lexical Analysis - Part 3
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 = { {...}, {...} , ..., {...} }
3
5 7
a a
{3}
NFA
a
{5,7}
DFA
(The names of the states is arbitrary and can be changed later, if desired.)
1 © Harry H. Porter, 2005
a
2 2
Lexical Analysis - Part 3
Algorithm: Convert NFA to DFA We’ll use... MoveNFA(S,a) #-Closure(s) #-Closure(S)
the transition function from NFA where s is a single state from NFA 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 {3,5,7} 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” The transition function from DFA To add an edge to the growing DFA...
MoveDFA(T,b)
Set MoveDFA(T,b) to S b
T
S
...where S and T are sets of NFA states © Harry H. Porter, 2005
3
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) =
© Harry H. Porter, 2005
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
4
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7}
2 4
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
© Harry H. Porter, 2005
5
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
© Harry H. Porter, 2005
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
6
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(A,a) =
MoveDFA(A,b) =
© Harry H. Porter, 2005
7
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = MoveDFA(A,b) =
© Harry H. Porter, 2005
8
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = MoveDFA(A,b) =
© Harry H. Porter, 2005
9
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} MoveDFA(A,b) =
© Harry H. Porter, 2005
10
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(A,b) =
© Harry H. Porter, 2005
11
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(A,b) = #-Closure (MoveNFA(A,b)) =
© Harry H. Porter, 2005
12
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(A,b) = #-Closure (MoveNFA(A,b)) = #-Closure ({5}) =
© Harry H. Porter, 2005
13
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(A,b) = #-Closure (MoveNFA(A,b)) = #-Closure ({5}) = {1,2,4,5,6,7} = C
© Harry H. Porter, 2005
14
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(A,b) = #-Closure (MoveNFA(A,b)) = #-Closure ({5}) = {1,2,4,5,6,7} = C
C
b
So far:
A
B
a
© Harry H. Porter, 2005
15
Lexical Analysis - Part 3
Example 0
#
#
1 #
Start state: #-Closure (0) = {0, 1, 2, 4, 7} = A
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(A,a) = #-Closure (MoveNFA(A,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(A,b) = #-Closure (MoveNFA(A,b)) = #-Closure ({5}) = {1,2,4,5,6,7} = C So far:
$ A
© Harry H. Porter, 2005
b a
C B
A is now done; mark it! B and C are unmarked. Let’s do B next... 16
Lexical Analysis - Part 3
Example 0
#
#
1 #
2 4
Process B = {1,2,3,4,6,7,8}
$ A
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
C
b
B
a
© Harry H. Porter, 2005
17
Lexical Analysis - Part 3
Example 0
#
#
1 #
Process B = {1,2,3,4,6,7,8}
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(B,a) =
$ A © Harry H. Porter, 2005
b a
C B 18
Lexical Analysis - Part 3
Example 0
#
#
1 #
2 4
Process B = {1,2,3,4,6,7,8}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) =
$ A
C
b
B
a
© Harry H. Porter, 2005
19
Lexical Analysis - Part 3
Example 0
#
#
1 #
Process B = {1,2,3,4,6,7,8}
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) =
$ A © Harry H. Porter, 2005
b a
C B 20
Lexical Analysis - Part 3
Example 0
#
#
1 #
2 4
Process B = {1,2,3,4,6,7,8}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B
$ A
C
b
B
a
© Harry H. Porter, 2005
21
Lexical Analysis - Part 3
Example 0
#
#
1 #
Process B = {1,2,3,4,6,7,8}
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B
$ A
b a
C B a
© Harry H. Porter, 2005
22
Lexical Analysis - Part 3
Example 0
#
#
1 #
2 4
Process B = {1,2,3,4,6,7,8}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(B,b) =
$ A
C
b
B
a
a © Harry H. Porter, 2005
23
Lexical Analysis - Part 3
Example 0
#
#
1 #
Process B = {1,2,3,4,6,7,8}
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(B,b) = #-Closure (MoveNFA(B,b)) =
$ A
b a
C B a
© Harry H. Porter, 2005
24
Lexical Analysis - Part 3
Example 0
#
#
1 #
2 4
Process B = {1,2,3,4,6,7,8}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(B,b) = #-Closure (MoveNFA(B,b)) = #-Closure ({5,9}) =
$ A
C
b
B
a
a © Harry H. Porter, 2005
25
Lexical Analysis - Part 3
Example 0
#
#
1 #
Process B = {1,2,3,4,6,7,8}
2 4
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(B,b) = #-Closure (MoveNFA(B,b)) = #-Closure ({5,9}) = {1,2,4,5,6,7,9} = D
$ A
b a
C B a
© Harry H. Porter, 2005
26
Lexical Analysis - Part 3
Example 0
2
#
#
1
4
#
Process B = {1,2,3,4,6,7,8}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(B,b) = #-Closure (MoveNFA(B,b)) = #-Closure ({5,9}) = {1,2,4,5,6,7,9} = D
$ A
C
b
B
a
D
b a
© Harry H. Porter, 2005
27
Lexical Analysis - Part 3
Example 0
2
#
#
1
4
#
Process B = {1,2,3,4,6,7,8}
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(B,a) = #-Closure (MoveNFA(B,a)) = #-Closure ({3,8}) = {1,2,3,4,6,7,8} = B MoveDFA(B,b) = #-Closure (MoveNFA(B,b)) = #-Closure ({5,9}) = {1,2,4,5,6,7,9} = D
$ A
b a
C B
$ b
D
a © Harry H. Porter, 2005
28
Lexical Analysis - Part 3
Example 0
2
#
#
1
4
#
Process C = {1,2,4,5,6,7}
$ A
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
C
b
B
a
$
D
b a
© Harry H. Porter, 2005
29
Lexical Analysis - Part 3
Example 0
2
#
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(C,a) = MoveDFA(C,b) =
$ A
b a
C B
$ b
D
a © Harry H. Porter, 2005
30
Lexical Analysis - Part 3
Example 0
2
#
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) =
$ A
C
b
B
a
$
D
b a
© Harry H. Porter, 2005
31
Lexical Analysis - Part 3
Example 0
2
#
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C
$ A
b a
C B
$ b
D
a © Harry H. Porter, 2005
32
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C
b
$ A
$
C
b
a
B
a
$
D
b a
© Harry H. Porter, 2005
33
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
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) = b
$ A
b a
C a
B
$ $ b
D
a © Harry H. Porter, 2005
34
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
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) = b
$ A
$
C
b
a
B
a
$
D
b a
© Harry H. Porter, 2005
35
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
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 b
$ A
b a
C a
B
$ $ b
D
a © Harry H. Porter, 2005
36
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
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 b
$ A
$
C
b
a
B
a
a
$
D
b
$
E
b
a © Harry H. Porter, 2005
37
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C Process E = {1,2,4,5,6,7,10}
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
MoveDFA(E,a) = MoveDFA(E,b) =
b
$ A
b a
C a
B
$ a
$ b
D
$ b
E
a © Harry H. Porter, 2005
38
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C Process E = {1,2,4,5,6,7,10}
Process D = {1,2,4,5,6,7,9}
MoveDFA(E,a) = {1,2,3,4,6,7,8} = B MoveDFA(E,b) =
MoveDFA(D,a) = {1,2,3,4,6,7,8} = B MoveDFA(D,b) = {1,2,4,5,6,7,10} = E b
$ A
$
C
b
a
B
a
a
$
D
b
$
E
b
a © Harry H. Porter, 2005
39
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3 5
%= {a,b}
#
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C Process E = {1,2,4,5,6,7,10}
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
MoveDFA(E,a) = {1,2,3,4,6,7,8} = B MoveDFA(E,b) = {1,2,4,5,6,7} = C
b
$ A
b a
C a
B
$ a
$ b
D
$ b
E
a © Harry H. Porter, 2005
40
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3
%= {a,b}
#
5
6
#
7
#
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C Process E = {1,2,4,5,6,7,10}
Process D = {1,2,4,5,6,7,9}
MoveDFA(E,a) = {1,2,3,4,6,7,8} = B MoveDFA(E,b) = {1,2,4,5,6,7} = C
MoveDFA(D,a) = {1,2,3,4,6,7,8} = B MoveDFA(D,b) = {1,2,4,5,6,7,10} = E b
$ A
C
b
a
B
a
b
$ a a
$
D
b
$
E
b
$
a © Harry H. Porter, 2005
41
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3
%= {a,b}
#
5
6 #
#
7
a
8
b
9
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C Process E = {1,2,4,5,6,7,10}
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 b
$ A
b a
C a
B
Final States in DFA? ...which state(s) contain 10?
b
$
MoveDFA(E,a) = {1,2,3,4,6,7,8} = B MoveDFA(E,b) = {1,2,4,5,6,7} = C
a a
$ b
D
$ b
E
$
a © Harry H. Porter, 2005
42
Lexical Analysis - Part 3
Example #
0
2
#
1
4
#
Process C = {1,2,4,5,6,7}
a
b
#
3
%= {a,b}
#
5
6
#
7
#
a
8
9
b
b
10
#
MoveDFA(C,a) = {1,2,3,4,6,7,8} = B MoveDFA(C,b) = {1,2,4,5,6,7} = C Process E = {1,2,4,5,6,7,10}
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 b
Final Result:
$ A
C
b
a
a
B
Final States in DFA? ...which state(s) contain 10?
b
$
MoveDFA(E,a) = {1,2,3,4,6,7,8} = B MoveDFA(E,b) = {1,2,4,5,6,7} = C
a a
$ b
D
$ b
E
$
a © Harry H. Porter, 2005
43
Lexical Analysis - Part 3
Algorithm: Convert NFA to DFA SDFA = {} Add #-Closure(s0) to SDFA as the start state Set the only state in SDFA to “unmarked” while SDFA contains an unmarked state do Let T be that unmarked state A set of NFA states Mark T for each a in % do Everywhere you could S = #-Closure(MoveNFA(T,a)) possibly get to on an a if S is not in SDFA already then Add S to SDFA (as an “unmarked” state) endIf Set MoveDFA(T,a) to S i.e, add an edge to the DFA... a
{t1,t2,...}
endFor T 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 © Harry H. Porter, 2005
{s1,s2,...}
S
44
Lexical Analysis - Part 3
Resulting DFA for (a|b)*abb b b
C b
A
a
a
a
B
a
D
b
E
b
a
Is it minimal?
© Harry H. Porter, 2005
45
Lexical Analysis - Part 3
Resulting DFA for (a|b)*abb b b
C b
A
a
a
a
B
a
D
b
E
b
a
Is it minimal?
A
© Harry H. Porter, 2005
a
B
b
D
b
E
46
Lexical Analysis - Part 3
Resulting DFA for (a|b)*abb b b
C b
A
a
a
a
B
a
D
b
E
b
a
Is it minimal?
A
B
a
D
b
E
b
© Harry H. Porter, 2005
47
Lexical Analysis - Part 3
Resulting DFA for (a|b)*abb b b
C b
A
a
a
a
B
a
D
b
E
b
a b
Is it minimal?
A
a
B
b
D
b
E
b
© Harry H. Porter, 2005
48
Lexical Analysis - Part 3
Resulting DFA for (a|b)*abb b b
C b
A
a
a
a
B
a
D
b
E
b
a b
Is it minimal?
a a
A
B
a
b
D
b
E
b
a
© Harry H. Porter, 2005
49
Lexical Analysis - Part 3
Resulting DFA for (a|b)*abb b b
C b
A
a
a
a
B
a
D
b
E
b
a b
Is it minimal?
Every Regular Set is recognized by a minimal DFA!
a a
A b
© Harry H. Porter, 2005
a
B
b
D
b
E
a
50
Lexical Analysis - Part 3
Resulting DFA for (a|b)*abb b b
C b
A
a
a
a
B
a
D
b
E
b
a b
Is it minimal?
Every Regular Set is recognized by a minimal DFA!
a a
A b
© Harry H. Porter, 2005
a
B a
b
D
b
E And it is unique, up to renaming of states
51
