Arden’s Theorem in Theory of Computation

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 an 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] 

Hence proved.

Thus, R = QP* is the unique solution of the equation R = Q + RP.

To understand this theorem, we will solve an example:

Example –

q1 = q1.0  + 
q2 = q1.1 + q2.0
q3 = q2.1 + q3.0 + q3.1 

Now,

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*.