Eliminating ε Transitions

NFA with ε can be converted to NFA without ε, and this NFA without ε can be converted to DFA. To do this, we will use a method, which can remove all the ε transition from given NFA. The method will be:

Find out all the ε transitions from each state from Q. That will be called as ε-closure{q1} where qi ∈ Q.
Then δ' transitions can be obtained. The δ' transitions mean a ε-closure on δ moves.
Repeat Step-2 for each input symbol and each state of given NFA.
Using the resultant states, the transition table for equivalent NFA without ε can be built.

Example:

Convert the following NFA with ε to NFA without ε.

Solutions: We will first obtain ε-closures of q0, q1 and q2 as follows:

ε-closure(q0) = {q0}
ε-closure(q1) = {q1, q2}
ε-closure(q2) = {q2}

Now the δ' transition on each input symbol is obtained as:

δ'(q0, a) = ε-closure(δ(δ^(q0, ε),a))
              = ε-closure(δ(ε-closure(q0),a))
              = ε-closure(δ(q0, a))
              = ε-closure(q1)
              = {q1, q2}

δ'(q0, b) = ε-closure(δ(δ^(q0, ε),b))
              = ε-closure(δ(ε-closure(q0),b))
              = ε-closure(δ(q0, b))
              = Ф

Now the δ' transition on q1 is obtained as:

δ'(q1, a) = ε-closure(δ(δ^(q1, ε),a))
              = ε-closure(δ(ε-closure(q1),a))
              = ε-closure(δ(q1, q2), a)
              = ε-closure(δ(q1, a) ∪ δ(q2, a))
              = ε-closure(Ф ∪ Ф)
              = Ф

δ'(q1, b) = ε-closure(δ(δ^(q1, ε),b))
              = ε-closure(δ(ε-closure(q1),b))
              = ε-closure(δ(q1, q2), b)
              = ε-closure(δ(q1, b) ∪ δ(q2, b))
              = ε-closure(Ф ∪ q2)
              = {q2}

The δ' transition on q2 is obtained as:

δ'(q2, a) = ε-closure(δ(δ^(q2, ε),a))
              = ε-closure(δ(ε-closure(q2),a))
              = ε-closure(δ(q2, a))
              = ε-closure(Ф)
              = Ф

δ'(q2, b) = ε-closure(δ(δ^(q2, ε),b))
              = ε-closure(δ(ε-closure(q2),b))
              = ε-closure(δ(q2, b))
              = ε-closure(q2)
              = {q2}

Now we will summarize all the computed δ' transitions:

δ'(q0, a) = {q0, q1}
δ'(q0, b) = Ф
δ'(q1, a) = Ф
δ'(q1, b) = {q2}
δ'(q2, a) = Ф
δ'(q2, b) = {q2}

The transition table can be:

States	a	b
→q0	{q1, q2}	Ф
*q1	Ф	{q2}
*q2	Ф	{q2}

State q1 and q2 become the final state as ε-closure of q1 and q2 contain the final state q2. The NFA can be shown by the following transition diagram: