Automata theory - RE to DFA Conversion

Convert RE to DFA
1
a
2 b
4
5 6
c
7
𝜀
𝜀
𝜀
8
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
Dstates
𝜀
Transition table of DFA
Given RE = (ab+c)*
Equivalent NFA-ε is:

Convert RE to DFA
𝜀
NFA States
DFA
State
Next State
a b c
Dstates
Transition table of DFA
Initial state of NFA is {9}
Ε-closure(9)={9,7,1,5,10} ------
Where A is the initial state of DFA
A

Mark state A
1
a
2 b
4
5 6
c
7
𝜀
𝜀
𝜀
8
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A   
Dstates
𝜀
Next need to find transition of (A,a) (A,b) (A,c)

Compute 𝜀-closure(move(A, a))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A
Dstates
𝜀
δ( A,a) = δ( (9,7,1,5,10),a) = {2}
=Ε-closure(2) = {2} ------ B

Compute 𝜀-closure(move(A, a))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B
{2} B
Dstates
𝜀
δ( A,a) = δ( (9,7,1,5,10),a) = {2}
Ε-closure(2) = {2} ------ B

Compute 𝜀-closure(move(A, b))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B -
{2} B
Dstates
𝜀

Compute 𝜀-closure(move(A, c))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B -
{2} B
Dstates
𝜀
δ( A,c) = δ( (9,7,1,5,10),c) = {6}
Ε-closure(6) = {6,8,10,7,1,5} ------ C


Mark move(A, c)) C
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B
{6,8,10,7,1,5} C
Dstates
𝜀

Compute 𝜀-closure(move(B, a))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B -
{6,8,10,7,1,5} C
Dstates
𝜀


Compute 𝜀-closure(move(B, b))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B -
{6,8,10,7,1,5} C
Dstates
𝜀
δ( B,b) = δ( (2,b) = {4}
Ε-closure(4) = {4,8,10,7,1,5} ------ D


Mark (move(B, b))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A  B - C
{2} B  - D
{6,8,10,7,1,5} C
{4,8,7,1,5,10} D
Dstates
𝜀

Mark - (move(B, c))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C
{4,8,7,1,5,10} D
Dstates
𝜀

Compute 𝜀-closure(move(C, a))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B
{4,8,7,1,5,10} D
Dstates
𝜀
B
δ( C,a) = δ( (6,8,10,7,1,5)),a) = {2}
Ε-closure(2) = {2} ------

Mark (move(C, b))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B -
{4,8,7,1,5,10} D
Dstates
𝜀

Compute 𝜀-closure(move(C, c))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B -
{4,8,7,1,5,10} D
Dstates
𝜀
δ( C,c) = δ( (6,8,10,7,1,5)),a) = {6}
Ε-closure(6) = {6,8,10,7,1,5} ------ C

Mark (move(C, c))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B - C
{4,8,7,1,5,10} D
Dstates
𝜀

Compute 𝜀-closure(move(D, a))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B - C
{4,8,7,1,5,10} D
Dstates
𝜀
δ( D,a) = δ( (6,8,10,7,1,5)),a) = {2}
Ε-closure(2) = {2} ------ B

Mark (move(D, a))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B - C
{4,8,7,1,5,10} D B
Dstates
𝜀

Mark (move(D, b))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B - C
{4,8,7,1,5,10} D B -
Dstates
𝜀

Compute 𝜀-closure(move(D, c))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B - C
{4,8,7,1,5,10} D B
Dstates
𝜀
C
( D,c) = δ( (4,8,7,1,5,10)),c) = {6}
Ε-closure(6) = {6,8,10,7,1,5} ------

Mark (move(D, c))
1
a
2 b
4
5 6
c
7
𝜀
𝜀
8
𝜀
𝜀
9 𝜀 𝜀
𝜀
10
NFA States
DFA
State
Next State
a b c
{9,7,1,5,10} A B - C
{2} B - D -
{6,8,10,7,1,5} C B - C
{4,8,7,1,5,10} D B - C
Dstates
𝜀

Resultant DFA is
a
B
c
a b
A a D
c
C
c
Transition table Transition diagram is

Automata theory - RE to DFA Conversion

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