CMSC 330 - Spring 2011 1

Finite Automaton: Example 1

0 0 1 0 1 1accepted

CMSC 330 - Spring 2011 2

Finite Automaton: Example 2

0 0 1 0 1 0not accepted

Reducing REs to NFAs

CMSC 330 - Spring 2011 3

ab

b

ε

a

b

✓

NFA for (ab|ba)*

CMSC 330 - Spring 2011 4

(ab|ba)*

b(ab|ba)* a(ab|ba)*

a b

✓

b a

NFA for (ab|aba)*

CMSC 330 - Spring 2011 5

(ab|aba)*

b(ab|aba)* ba(ab|aba)*

a a

✓

b

a

a(ab|aba)*

b

6

NFA DFA Example 1

• r0 = {S1}

• R = {r0} = { {S1} }

• r R = {S1}

• move({S1}, a} = {S2,S3} R = R ∪ {{S2,S3}} = { {S1}, {S2,S3} } = ∪ {<{S1}, a, {S2,S3}>}

• move({S1},b} = Ø R = R ∪ {Ø}

= { Ø, {S1}, {S2,S3} } = ∪ {<{S1}, b, Ø>}

bS1 S2 S3

a

a

a{S2,S3}{S1}

NFA

DFA

∅

b

7

NFA DFA Example 1 (cont.)

• R = { {S1}, {S2,S3}, Ø }

• r R = {S2,S3}

• move({S2,S3},a} = Ø

• move({S2,S3},b} = {S3} R = R ∪ {{S3} = { {S1}, {S2,S3} }, Ø, {S3} } = ∪ {<{S2,S3},a, Ø >, <{S2,S3}, b, {S3}>}

bS1 S2 S3

a

a

a b{S3}{S1}

NFA

DFA

∅

b{S2,S3}

a

8

NFA DFA Example 1 (cont.)

• R = { {S1}, {S2,S3}, Ø, {S3} }

• r R = Ø

• move(Ø,a} = Ø

• move(Ø,b} = Ø R = { {S1}, {S2,S3} }, Ø, {S3} } = ∪ {< Ø,a,Ø >, <Ø,b, Ø>}

bS1 S2 S3

a

a

a b{S1}

NFA

DFA

∅

b{S2,S3}

a

a b

{S3}

9

NFA DFA Example 1 (cont.)

• R = { {S1}, {S2,S3}, Ø, {S3} }

• r R = {S3}

• move({S3},a} = Ø

• move({S3},b} = Ø R = { {S1}, {S2,S3} }, Ø, {S3} } = ∪ {<{S3},a, Ø >, <{S3}, b, Ø>}

• Fd = {{S2,S3}, {S3}} Since S3 Fn

• Done!

bS1 S2 S3

a

a

NFA

DFA

a b{S1}

∅

b{S2,S3}

a

a b

{S3}

a,b