Arden’s theorem helps in checking the equivalence of two regular expressions.
Let, P and Q be two regular expressions over the input set Σ. The regular expression R is given as follows −
This has a unique solution as R=QP*.
Let, P and Q be the two regular expressions over the input string Σ.
If P does not contain ε then there exists R such that
R= Q+RP-----------------------equation 1
We will replace R by QP* in equation 1
Consider Right hand side (R.H.S) of equation 1
=QP* since R*R=R* according to identity
Thus R=QP* is proved.
To prove that R=QP* is a unique solution, we will now replace left hand side (L.H.S) of eq 1 by Q+RP
Then it becomes, Q+RP
But again, R can be replaced by Q+RP
Q+RP = Q+(Q+RP)P
Again, replace R by Q+RP
Thus, if we go on replacing R by Q+RP then we get,
From equation 1
R = Q(ε+P+P2+…….+Pi)+RPi+1 ---------------equation 2
Consider equation 2
Let w be a string of length i
In RPi+1 has no string of less than i+1 length.
Hence, it is proved that
R=Q+RP has a unique solution