Prove a language is not regular without pumping lemma

Question

How can you prove that $L=\{a^n b^{2n} \}$ is not regular without the use of pumping lemma?

Answer 1

There are several ways to prove a language is not regular:

• the pumping lemma is one of them;
• you can use closure properties of regular languages. For example, if $L_1$ and $L_2$ are regular then $L = L_1 \cap L_2$ must be regular. That means that if $L$ is not regular, then either $L_1$ or $L_2$ is not regular;
• you can use the Myhill-Nerode theorem;
• you can suppose a language is recognized by a DFA, and conclude to a contradiction (the DFA must have at least $n$ states for all $n\in\mathbb{N}$, it must recognize words not in the language, …)