Conversion from NFA to DFA

In this section, we will discuss the method of converting NFA to its equivalent DFA. In NFA, when a specific input is given to the current state, the machine goes to multiple states. It can have zero, one or more than one move on a given input symbol. On the other hand, in DFA, when a specific input is given to the current state, the machine goes to only one state. DFA has only one move on a given input symbol.

Let, M = (Q, ∑, δ, q0, F) is an NFA which accepts the language L(M). There should be equivalent DFA denoted by M' = (Q', ∑', q0', δ', F') such that L(M) = L(M').

Steps for converting NFA to DFA:

Step 1: Initially Q' = ϕ

Step 2: Add q0 of NFA to Q'. Then find the transitions from this start state.

Step 3: In Q', find the possible set of states for each input symbol. If this set of states is not in Q', then add it to Q'.

Step 4: In DFA, the final state will be all the states which contain F(final states of NFA)

Example 1:

Convert the given NFA to DFA.

Solution: For the given transition diagram we will first construct the transition table.

Now we will obtain δ' transition for state q0.

δ'([q0], 0) = [q0]
δ'([q0], 1) = [q1]

The δ' transition for state q1 is obtained as:

δ'([q1], 0) = [q1, q2]       (new state generated)
δ'([q1], 1) = [q1]

The δ' transition for state q2 is obtained as:

δ'([q2], 0) = [q2]
δ'([q2], 1) = [q1, q2]

Now we will obtain δ' transition on [q1, q2].

δ'([q1, q2], 0) = δ(q1, 0) ∪ δ(q2, 0)
                      = {q1, q2} ∪ {q2}
                      = [q1, q2]
δ'([q1, q2], 1) = δ(q1, 1) ∪ δ(q2, 1)
                      = {q1} ∪ {q1, q2}
                      = {q1, q2}
                      = [q1, q2]

The state [q1, q2] is the final state as well because it contains a final state q2. The transition table for the constructed DFA will be:

The Transition diagram will be:

The state q2 can be eliminated because q2 is an unreachable state.

Example 2:

Convert the given NFA to DFA.

Solution: For the given transition diagram we will first construct the transition table.

Now we will obtain δ' transition for state q0.

δ'([q0], 0) = {q0, q1}
               = [q0, q1]       (new state generated)
δ'([q0], 1) = {q1} = [q1]

The δ' transition for state q1 is obtained as:

δ'([q1], 0) = ϕ
δ'([q1], 1) = [q0, q1]

Now we will obtain δ' transition on [q0, q1].

δ'([q0, q1], 0) = δ(q0, 0) ∪ δ(q1, 0)
                      = {q0, q1} ∪ ϕ
                      = {q0, q1}
                      = [q0, q1]

Similarly,

δ'([q0, q1], 1) = δ(q0, 1) ∪ δ(q1, 1)
                      = {q1} ∪ {q0, q1}
                      = {q0, q1}
                      = [q0, q1]

As in the given NFA, q1 is a final state, then in DFA wherever, q1 exists that state becomes a final state. Hence in the DFA, final states are [q1] and [q0, q1]. Therefore set of final states F = {[q1], [q0, q1]}.

The transition table for the constructed DFA will be:

The Transition diagram will be:

Even we can change the name of the states of DFA.

Suppose

A = [q0]
B = [q1]
C = [q0, q1]

With these new names the DFA will be as follows: