Is this considered a regular expression?

Question

I need to design an NFA that recognizes the language $L$, with alphabet $\{a,b\}$, that accepts strings that:

• have length at least 4
• two letters before the final letter of the string there is always a $b$

For example the language accepts $abba$ but neither $aba$ nor $abbaba$. I designed this:

If I wanted to provide a regular expression also I was wondering if I could write :
$(a|b)^*b(a|b)^2$
I have never seen an exponent as $2$ that's why I am concerned.

Answer 1

Denoting by $A$ the alphabet, your language is $A^*AbA^2$. If $L$ is any language, $L^n$ is defined by induction as follows: $L^0 = \{1\}$ (where $1$ denotes the empty word, which is the identity of the free monoid $A^*$) and $L^{n+1} = L^nL$. Thus $L^2 = LL$.

You can remove state $1$ in your non-deterministic automaton.