L = {$xy ∈ \{a, b\}^∗ : |x| = |y|$ and x contains the substring aa}

I am trying to prove that this language is not regular using MyHill-Nerode theorem but I am unable to find equivelance classes of this language. Thanks.

  • 4
    You don't need to find the equivalence classes of the language. It suffices to find infinitely many words which are pairwise distinguishable. – Yuval Filmus Mar 23 at 11:42
up vote 3 down vote accepted

Consider two words $u_i = b^i aa, u_j = b^j aa \in \{a, b\}^\ast$ with $i < j$. We have that $u_i \not\sim_L u_j$ because if you concatenate the word $v = b^{i+2}$ to both words we obtain $$u_i \cdot v = b^i aa \cdot b^{i+2} \in L \text{, but } u_j \cdot v = b^j aa \cdot b^{i+2} \notin L.$$ Thus, $v$ separates $u_i$ and $u_j$ for $L$. Since we can find infinitely many $i, j \in \mathbb{N}$ with $i < j$ we have also infinitely many Myhill-Nerode-equivalence classes.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.