Arden’s theorem state that:
“If P and Q are two regular expressions over , and if P does not contain , then the following equation in R given by R = Q + RP has a unique solution i.e., R = QP*.”
That means, whenever we get any equation in the form of R = Q + RP, then we can directly replaced by R = QP*. So, here first we will prove that R = QP* is the solution of this equation and then we will also prove that it is the unique solution of this equation.
Let’s start by taking this equation as equation (i)
R = Q + RP ......(i)
Now, replacing R by R = QP*, we get,
R = Q + QP*P
Taking Q as common,
R = Q( + P*P) R = QP*
(As we know that + R*R = R*). Hence proved.
Thus, R = QP* is the solution of the equation R = Q + RP.
Now, we have to prove that this is the only solution of this equation. Let me take this equation again:
R = Q + RP
Now, replace R by R = Q + RP,
R = Q + (Q + RP)P = Q + QP + R
Again, replace R by R = Q + RP:-
R = Q + QP + (Q + RP) = Q + QP + Q + R = ... = ... = Q + QP + Q + .. + Q + R
Now, replace R by R = QP*, we get,
R = Q + QP + Q + .. + Q + QP*
Taking Q as common,
R = Q( + P + + .. + + P*) = QP* [As + P + + .. + + P* represent the closure of P]
Thus, R = QP* is the unique solution of the equation R = Q + RP.
To understand this theorem, we will solve an example:
q1 = q1.0 + q2 = q1.1 + q2.0 q3 = q2.1 + q3.0 + q3.1
q1 = + q1.0 q1 = .0* [By Arden's theorem] q1 = 0* [R = R] .'. q2 = 0*1 +q2.0 q2 = 0*10*
[Applying Arden’s theorem]. Hence, the value of q2 is 0*10*.
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