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Proving that L is not regular by showing that $\equiv_L$ has infinite index.
$\Sigma$ = {a}, L = {$a^{3^n} : n \geq$ 0}
My ideas:
theorem of Myhill-Nerode: L $\in$REG $\Leftrightarrow$ $\equiv_L$ has finite index
We show: $\equiv_L$ has infinte equivalence classes.
$a^{3^i}$ $\not\equiv$ $a^{3^{i+k}}$, $k>0$
-> equivalence classes: [a], [aaa], [$a^3$]...
Every $a^{3^i}$ is a different equivalence class. Therefore $\equiv_L$ has an infinite amount of equivalence classes $\rightarrow$ L is not regular. qed.

How to show that the classes are different?

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  • $\begingroup$ I dont know how you say, that the equivalence classes are different. So how to show that? $\endgroup$ – joee Nov 12 '18 at 11:20
  • $\begingroup$ Okay, that's a fair question. Hint: you're trying to show inequality of sets. $\endgroup$ – Raphael Nov 12 '18 at 13:27
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Let me prove a stronger claim: every two words over $\{a\}$ are inequivalent with respect to your language. What this means is that for every $i \neq j$, there is a word $w \in a^*$ such that either $a^i w \in L$ and $a^j w \notin L$, or vice versa. Since $w \in a^*$, in fact $w = a^k$ for some $k$, and so we need to prove the following:

If $i \neq j$ are natural numbers then there exists a natural number $k$ such that exactly one of $i+k,j+k$ is a power of 3.

Let $a_i$ be the smallest power of 3 which is at least $i$, and define $a_j$ similarly. If $a_i \neq a_j$, say $a_i < a_j$, then we can take $k = a_i$. So suppose $a_i = a_j = a$. Without loss of generality, $i < j$, and so $i+a < j+a$. The next power of 3 after $i+a$ is $3(i+a)$, and after $j+a$ is $3(j+a)$. Let $k = 2i+3a$. Then $i + k$ is a power of 3 by definition, but $j+a < j + k < 3(j+a)$ is not a power of 3.

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