Converting an NFA into an FSA

Proving LNFA is a subset of LFSA.

Purpose

• This presentation presents an example execution of the algorithm which takes as input an NFA (without -transitions) and produces as output an equivalent FSA

• Algorithm Specification

– Input: NFA M1 (without -transitions)

– Output: FSA M2 such that L(M2) = L(M1)

Input NFA M1

a

b

a,b

b b

a a

1

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6

Initialization

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a,b

b b

a a

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a b{1}

Initial State of FSA M2: {1}

Expand State {1}

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a,b

b b

a a

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6

a b{1} {2} {3}

Add States {2} and {3}

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a,b

b b

a a

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a b{1} {2} {3}{2}{3}

Expand State {2}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3}

Add State {}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3}{}

Expand State {3}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{}

Add States {3,4} and {3,5}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{}

{3,4}{3,5}

Expand State {}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4}{3,5}

No New States Added

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4}{3,5}

Expand State {3,4}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5}

Add State {3,4,6}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5}

{3,4,6}

Expand State {3,5}

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b

a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6}

Add State {3,5,6}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6}{3,5,6}

Expand State {3,4,6}

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a,b

b b

a a

1

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6} {3,4,6} {3,5}{3,5,6}

No New States Added

a

b

a,b

b b

a a

1

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6

a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6} {3,4,6} {3,5}{3,5,6}

Expand State {3,5,6}

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6} {3,4,6} {3,5}{3,5,6} {3,4} {3,5,6}

No New States Added

a

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a,b

b b

a a

1

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6} {3,4,6} {3,5}{3,5,6} {3,4} {3,5,6}

Identify Accepting States

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6} {3,4,6} {3,5}{3,5,6} {3,4} {3,5,6}

Accepting states of FSA M2: {3,4,6} and {3,5,6}

Transition Diagram

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a,b

b b

a a

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a b{1} {2} {3}{2} {} {}{3} {3,4} {3,5}{} {} {}

{3,4} {3,4,6} {3,5}{3,5} {3,4} {3,5,6}

{3,4,6} {3,4,6} {3,5}{3,5,6} {3,4} {3,5,6}

Accepting states of FSA M2: {3,4,6} and {3,5,6}

{1} {2} {}

{3}

{3,4}

{3,4,6} {3,5,6}

{3,5}

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a,b

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FSA M2

Summary

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a a

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{1} {2} {}

{3}

{3,4}

{3,4,6} {3,5,6}

{3,5}

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a,b

a

a

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ab

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Original NFA M1 Equivalent FSA M2