# How to apply Arden's theory to determine a regular expression

If $$P=ab$$ and $$Q=a^*$$, how do I use Arden's theorem to determine the regular expression $$R$$. I'm not sure if I am supposed to just substitute the values of $$P$$ and $$Q$$ in the equation $$R= Q + RP$$. Also how would I use that to check that $$R$$ satisfies Arden's equation.

• It's not at all clear what you're asking here. Questions should be reasonably self-contained. Could you please explain how R relates to P and Q, and perhaps even give the statement of Arden's theorem (since it's very short)? Thanks. – Pseudonym Sep 6 at 23:11

Arden's theorem states that $$A^*\,B$$ is the least fixed point of the equation:
$$X = A\,X\,\cup B$$
and that $$A\,B^*$$ is the least fixed point of the equation:
$$X = X\,B\,\cup A$$
In your case, $$R = Q\,P^* = a^* (a b)^*$$.