Explain the complementation process in DFA

The complementation process in the deterministic finite automata (DFA) is explained below −

Let’s take a DFA which is defined by (Q, Σ, δ,q0,F) and it accepts the language L1. Now, the DFA that accepts the language L2, where L2 = ̅L1, is defined as follows −

         (Q, Σ, δ,q0,Q-F)

The complement of a DFA is obtained by making the non-final states as final states and final states as non-final states.

The language which is accepted by the complemented DFA L2 is the complement of language L1.

Examples

Let’s consider some examples to get the clarity on the complementation process of DFA.

Example 1

Consider two languages L1 and L2.

In L1 all strings start with ‘a’ over an alphabet {a,b}

L1={a,ab,aa,aba,aab,aaa,……}

In L2 all strings not starting with ‘a’ over an alphabet {a,b}

L2={€, b,ba,bab,baa,bba,……}

Here we can observe that the two languages are in the form of −

L2 = ̅L1

The transition diagram of DFA for L1 that accepts the set of all strings over {a, b} starting with ‘a’ is given below −

q0 on ‘a’ goes to q1 which is a final state and generates the strings that starts with letter ‘a’

{a,ab,aab,abb,aaab,……}

Where q2 is the dead state.

Construct DFA for the complement

Now, we can construct DFA for the complement simply by interchanging the final and non-final states.

This refers to the change of non-final states as final states and final states as non-final states.

The state transition diagram after constructing DFA for the complement is as follows −

The above transition diagram is the complemented DFA which accepts the strings that are not starting with ‘a’.

q0 on ‘a’ goes to q1 which is a dead state so, it is cleared the above state transition DFA will not generate the strings starting with ‘a’.