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NFA to DFA Conversion. Rabin and Scott (1959). Removing Nondeterminism. By simulating all moves of an NFA- λ in parallel using a DFA. λ -closure of a state is the set of states reachable using only the λ -transitions. . NFA- λ. p2. λ. p1. a. p3. q1. λ. λ. p5. λ. q2. a. p4.

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NFA to DFA Conversion

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NFA to DFA Conversion

Rabin and Scott (1959)

L12NFA2DFA


Removing Nondeterminism

  • By simulating all moves of an NFA-λin parallel using a DFA.

  • λ-closure of a state is the set of states reachable using only the λ-transitions.

L12NFA2DFA


NFA-λ

p2

λ

p1

a

p3

q1

λ

λ

p5

λ

q2

a

p4

L12NFA2DFA


L12NFA2DFA


Transition Function

L12NFA2DFA


L12NFA2DFA


Equivalence Construction

  • Given NFA-λ M, construct a DFA DM such that L(M) = L(DM).

  • Observe that

    • A node of the DFA = Set of nodes of NFA-λ

    • Transition of the DFA = Transition among set of nodes of NFA- λ

L12NFA2DFA


{q1,…,qn}

{p1,…,pm}

a

For every pi, there exists qj such that

L12NFA2DFA


Start state of DFA =

Final/Accepting state of DFA =

All subsets of states of NFA-λ

that contain an accepting state

of the NFA-λ

Dead state of DFA =

L12NFA2DFA


Example

a

b

q1

a

q0

a+c*b*

λ

a

q2

c

L12NFA2DFA


{q0}

{q0}

{q0,q1,q2}

a

b

c

f

L12NFA2DFA


{q0}

{q0,q1,q2}

a

b

c

f

a

b

{q1}

c

{q1,q2}

a,b,c

L12NFA2DFA


b

{q1}

a,c

f

c

{q1}

b

{q1,q2}

a

f

L12NFA2DFA


Equivalent DFA

a

a

{q0}

{q0,q1,q2}

c

c

{q1,q2}

b

b,c

b

b

a,c

{q1}

f

a

a,b,c

L12NFA2DFA


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