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December 12th, 2015, 01:57 AM   #1
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Automata question

The question:
Let it be $L$ a regular language.
few definitions:
$p(L)$-the minimal natural number so that $L$ fulfill the pumping lemma.
$n(L)$- minimal NFA that accepts $L$.
$m(L)$- $Rank(L)$, the number of equivalence classes in $L$.

Let it be integer $k>0$. find an example for a language $L$ so that: $p=n=m=k$.

My attempt:
At start, thought of $L=\{w: |w|\bmod k=0\}$, $\Sigma=\{ a\}$.
But then I realized that $p<k$.
And also- for every $L$ that I choose here, I don't know how exactly to prove that the $NFA$ which accepts it is minimal. I know how to do this only on $NFA$'s with one or two states, but not on $NFA$'s with unknown number of states.
How can I solve this?
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