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$\{a^{2n}b^{n+k+1}a^k ∈ \{a, b\}^∗ \mid n \ge 0, k \ge 0\}$

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3
  • $\begingroup$ Please tell us what you have tried. $\endgroup$
    – Nobody
    Aug 9, 2023 at 1:58
  • $\begingroup$ This is unreadable. Use Matjax syntax. $\endgroup$
    – user16034
    Aug 10, 2023 at 16:14
  • $\begingroup$ My -1 for showing no effort, not even adapting the title of the exercise sheet. $\endgroup$
    – user16034
    Aug 10, 2023 at 16:20

1 Answer 1

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$L = \{ a^{2n} b^{n+k+1} a^k \mid n,k \ge 0\}$ is not a regular language.

Suppose towards a contradiction that $L$ is regular and let $p$ be its pumping length. Consider the word $b^{p} a^{p-1} \in L$. By the pumping lemma there is some positive integer $i \le p$ such that, for every $h \ge 0$: $$b^{p-i} b^{hi} a^{p-1} \in L.$$ Choosing $h=0$ we obtain: $$b^{p-i} a^{p-1} \in L,$$ which provides the sought contradiction.

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