Construct the minimum DFA for any given finite automata.

Problem

Construct a minimum state DFA for the following automata −

Solution

We first construct a transition table for the given finite automata −

Q={q0,q1,q2,q3,q4,q5,q6,q7}

Q01={q2} and Q02={q0,q1,q2,q3,q4,q5,q6,q7}

S0={{q2} {q0,q1,q2,q3,q4,q5,q6,q7}}

Consider the set {q0,q1,q2,q3,q4,q5,q6,q7}

{q2} {q0,q1,q3,q5,q6,q7}

{q2} {q0,q4,q6} {q1,q3,q5,q7}

{q2} {q0,q4} {q6} {q1,q3,q5,q7}

{q2}{q0,q4}{q6}{q1,q7}{q3,q5}

The minimized state is as follows −

M¹=(Q¹, Σ, δ¹,q0¹,F¹)

Q¹= {[q2],[q0,q4],[q6],[q1,q7],[q3,q5]}

qo¹= {[q0,q4]}

F¹= {[q2]}

Transition Table

Now the transition table is as follows −

The transition diagram is as follows −