From what I understood, a L is regular if you can draw a FSA or a Regex. For the L={$a^n b^2 a^n | n≥0$}, I've written out a Finite State Automata and a Regex. I'm not sure if this is correct, so can someone verify this for me?
$ stands for epsilon by the way.
Regex: aba
Thank you!
A good rule of thumb to determine whether a language is regular or not is if it requires counting. To form a valid string, your automaton would have to count the number of $a$'s it is adding after the two $b$'s, so that it matches the same number of $a$'s at the start. That is why it is not regular, because FSAs/regular expressions cannot count.
As mentioned in the comments, you can verify that this language is not regular formally using the pumping lemma.
The next level up from FSAs are push-down automata. A push-down automaton can (sort of) count, since it would use a stack to keep track of the number of $a$'s. This means that it is at least a deterministic context-free language.