Transcript NFA to DFA Conversion Rabin and Scott (1959) Prasad L12NFA2DFA

NFA to DFA Conversion
Rabin and Scott (1959)
Prasad
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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.
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NFA-λ
p2
λ
a
p1
λ
q1
p3
λ
λ
q2
p5
a
p4
t ( q1, a )  { p1, p 2, p3, p 4, p5}
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q1   - closure ( q1)
q2   - closure ( q1)
p3   - closure ( p1)
 - closure (U{qi })

i
U - closure (q )
i
i
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Transition Function
t : Q    Powerset (Q )
t ( qi , a ) 
q j  -closure( qi )
t ( q1 , a ) 
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 - closure ( ( q
U

U - closure ( (q
q j { q1 , q 2 }
L12NFA2DFA
j
j
, a ))
, a ))
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t ( q1 , a )   - closure ( ( q1 , a ))
  - closure ( ( q2 , a ))
t (q1, a)   - closure ( p1 )   - closure ( p4 )
t ( q1 , a )  { p1 , p2 , p3}  { p4 , p5}
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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
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a
{q1,…,qn}
{p1,…,pm}
For every pi, there exists qj such that

 (q j ,  * a*)  pi
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Start state of DFA =
 - closure ({q0})
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 =
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Example
a
b
q1
a
q0
a+c*b*
λ
a
q2
c
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{q0}
a
{q0}
b
c
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{q0,q1,q2}
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a
a
{q0}
{q0,q1,q2}
{q1}
c
b
c
b

{q1,q2}
a,b,c
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b
{q1}
a,c

c
b
{q1,q2}
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{q1}

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Equivalent DFA
a
a
{q0}
{q0,q1,q2}
b
b,c

a,c
b
{q1}
c
c
{q1,q2}
b
a
a,b,c
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