I'm trying to prove that the set of even-length strings with the two middle symbols being equal cannot be accepted by finite automata. I can explain why it cannot be accepted intuitively, but I'm having trouble with the proof. Our symbols are {a, b}.
I allowed L = $\{(ab)^{*{\frac{n}{2} - 1}} aa (ab)^{*{\frac{n}{2} - 1}}\}$. I know the format of the language is wonky, and will be talking to my professor about it tomorrow. For the proof, I allowed $\frac{n}{2} - 1$ to be the combination of symbols before and after the two elements. So, using the Pumping Lemma's condition that |uv|≤ n, I allowed $u = \frac{n}{2} -1$ and $v = n^2$ (for aa); this is obviously greater than n, but I'm having trouble understanding how to choose $u$ and $v$. Is my assignment for these parameters correct?