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My language is the repetition of 0 to a length that's a power of 2:

$L = \{ 0^k \mid k=2^n, n \geq 1 \}$

I want to know how to use the Myhill-Nerode theorem to show that this language is not regular.

This is my first attempt at doing this although I am confident that I am wrong:

$j, p = 2^h$ for 2 distinct values of $h$, $h \in \mathbb{N}$:

$a = 0^{j/2}$

$b = 0^{p/2}$

$c = 0^{j/2}$

$ac = 0^{j/2}0^{j/2} = 0^j$ is in my language since $j$ is of the form $2^n$

$bc = 0^{p/2}0^{j/2}$ is not guaranteed to be in my language for every $p$ and $j$, since $j \ne p$

Thus my language must not be regular since $ac$ is in it but $bc$ is not.

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If you want to use Myhill-Nerode to show that a language is not regular, you need to prove that there are infinitely many equivalence classes.

Basically here, it suffices to prove that each $0^j$ is in a different class.

For instance $0^9$ and $0^{10}$ are not in the same class, because $0^90^7\in L$ while $0^{10}0^7\notin L$. Try to write the general proof for any $0^i$ and $0^j$ with $i\neq j$.

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