converting DFA to regular expression using Transitive closure method

Question

The following example shows a simple DFA with one accepting state q2:

Based on the R(i,j,k) Algorithm shown above i want to convert this DFA to regular expression, unfortunately i can't find a good definition of the K. My question is what does the K mean? is it the number of states (in this case 3) or something else?

Answer 1

Then, we solve those equations to get the equation for qi in terms of α ij and that expression is the required solution, where qi is a final state. It is shown below −

q 1= qa + q3a + € ( € move is because q1is the initial state)

q2 = q1b + q2b + q3b

q3 = q2a

Now, we will solve these equations, as shown below−

. q2 = q1b + q2b + q3b

= q1b + q2b + (q2a)b (Substituting value of q3)

= q1b + q2(b + ab)

= q1b (b + ab) * (Applying Arden's Theorem)

. q1 = q1a + q3a + €

= q1a + q2aa + € (Substituting value of q3)

= q1a + q1b(b + ab*)aa + € (Substituting value of q2)

= q1(a + b(b + ab) *aa) + €

= € (a+ b(b + ab) *aa) *

= (a + b(b + ab) *aa) *

Hence, the regular expression is (a + b(b + ab) *aa) *.