Summary
Highlights
This lecture explains how to design regular expressions for DFAs that have multiple final states, unlike previous examples with a single final state. The method involves finding regular expressions for each final state and then taking their union.
The first step is to write down the equations for each state (q1, q2, q3). For q1, the equation is Epsilon + q1_0. For q2, it is q1_1 + q2_1. For q3, it's q2_0 + q3_0 + q3_1. These equations represent the incoming transitions to each state.
We begin by finding the regular expression for final state q1. The equation for q1 (Epsilon + q1_0) is in the form R = Q + RP. Using Arden's Theorem (R = QP*), q1 becomes Epsilon * 0*, which simplifies to 0*.
Next, we find the regular expression for the second final state, q2. The equation for q2 is q1_1 + q2_1. By substituting the previously found value of q1 (0*), the equation becomes 0*1 + q2_1. This is also in the form R = Q + RP, where R = q2, Q = 0*1, and P = 1. Applying Arden's Theorem, q2 evaluates to 0*11*.
Since there are two final states (q1 and q2), the overall regular expression for the DFA is the union of their individual regular expressions. This gives 0* + 0*11*. The expression can be simplified by factoring out 0*, resulting in 0*(Epsilon + 11*). Using the identity Epsilon + RR* = R*, this further simplifies to 0*1*.
In conclusion, to design a regular expression for a DFA with multiple final states, you must find the regular expression for each final state independently and then perform their union. This combined regular expression represents the language accepted by the DFA.