Arden's TheoremThe Arden's Theorem is useful for checking the equivalence of two regular expressions as well as in the conversion of DFA to a regular expression. Let us see its use in the conversion of DFA to a regular expression. Following algorithm is used to build the regular expression form given DFA. 1. Let q_{1} be the initial state. 2. There are q_{2}, q_{3}, q_{4} ....q_{n} number of states. The final state may be some q_{j} where j<= n. 3. Let α_{ji} represents the transition from q_{j} to q_{i}. 4. Calculate q_{i} such that q_{i} = α_{ji} * q_{j} If q_{j} is a start state then we have: q_{i} = α_{ji} * q_{j} + ε 5. Similarly, compute the final state which ultimately gives the regular expression 'r'. Example:Construct the regular expression for the given DFA Solution: Let us write down the equations q1 = q1 0 + ε Since q1 is the start state, so ε will be added, and the input 0 is coming to q1 from q1 hence we write Similarly, q2 = q1 1 + q2 1 q3 = q2 0 + q3 (0+1) Since the final states are q1 and q2, we are interested in solving q1 and q2 only. Let us see q1 first q1 = q1 0 + ε We can rewrite it as q1 = ε + q1 0 Which is similar to R = Q + RP, and gets reduced to R = OP*. Assuming R = q1, Q = ε, P = 0 We get q1 = ε.(0)* q1 = 0* (ε.R*= R*) Substituting the value into q2, we will get q2 = 0* 1 + q2 1 q2 = 0* 1 (1)* (R = Q + RP → Q P*) The regular expression is given by r = q1 + q2 = 0^{*} + 0^{*} 1.1^{*} r = 0^{*} + 0^{*} 1^{+} (1.1^{*} = 1^{+})
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