I have a homework about find the pumping lemma in Context Free Language. The last one I couldn't solve:
$L = \{a^i \$ a^{3i} \$ a^{5i} \mid i \in \mathbb{N} \}$
What does the dollar symbol mean in this language?
How can I use pumping lemma in this example?
Thank you in advance.
On
In theory there are several cases, but they all work exactly the same way and can be handled at once.
Start with the word $s=a^p\$a^{3p}\$a^{5p}$, where $p$ is the pumping length. The pumping lemma gives you a decomposition $s=uvwxy$ such that $|vwx|\le p$, $|vx|\ge 1$, and $uv^kwx^ky\in L$ for each $k\ge 0$. Because $|vwx|\le p$, the string $uvw$ can intersect at most two of the blocks of $a$s; use this fact to show easily that pumping $s$ takes you out of $L$.
The dollar symbol here acts as a delimiter symbol, it separates the three substrings of $a$s.