Giving a regular expression for a situation of a question

Question

I have the following question regarding a situation given and we are then required to derive a regular expression out of it.

What i have to do now i to give a regular expression for this language described by the knob movement from A,B and C. Because i have to go through B if i were to traverse to C, and vice versa, i know that B must be present as part of the expression.

Here's what i attempted:

ABC U ABA U BAB U BCB U CBA U CBC

Simplifying it:

AB(C U A) U B(AB U CB) U CB(A U C)

I'm not entirely sure if I'm doing it right as I find out that I'm brute forcing. I'm terrible at deriving regular expressions so I'll appreciate some corrections.

Answer 1

Here's a rough sketch: let $S_a, S_b, S_c$ be the set of valid strings of the language starting with $a, b, c$ respectively. Observe that:

$\qquad S_b = ((a^* + c^*)b)^*$

Furthermore:

$\qquad S_a = aS_a + bS_b$

and symmetrically for $S_c$. Therefore we may replace $S_b$ in the latter, obtaining a "recursive" equation solvable by Arden's lemma.