I was studing Pumping lemma and I saw wondering and confusing definition,"Pumping length".
Here is my problems:
Question 1:
Is pumping length = number of states in DFA?
Question 2:
If any string can be divided into $\{xy^iz$ | $ i≥ 0\}$ , why $\vert xy\vert ≤$ Pumping length ?
I know Pigeonhole principle and how it used to epxlain we have a loop in DFA.
But I dont know how to use Pigeonhole principle to proof $\vert xy\vert ≤$ pumping length .
I tried but gather nothing:
I also checked this links but nothing was clear to me:
First link
Second one
Thanks for any kind of help ;)
On your first question: This one is a bit ill-posed because you don't specify what (D)FA you are talking about. The PL only says that for every regular $L$, there exists some integer $p$ such that all words in $L$ of length at least $p$ can be pumped, i.e. it doesn't uniquely specify such a $p$ (of course one could fix that by requiring $p$ to be minimal).
If we take a look at the proof of the PL, we see that $p$ is chosen to be the number of states of a DFA recognizing $L$, so in that sense the answer to your question is yes, but as I laid out, the "pumping length" is really referring just to the specifically chosen $p$.
On the second question: This is usually stated as $|xy| \leq p$. The idea of the PL is that the run of our DFA on some pumpable word can be segmented into three parts: One that leads to a loop ($x$), then the loop itself ($y$), and finally the part that leads from that loop to an accepting state ($z$). The proof argues that such a decomposition exists for all words of length at least $p$ by (as you said) employing the pidgeonhole principle. Now, this same argument tells us that a loop must already occur after at most $p$ symbols and hence, $|xy|$ is at most $p$.
I want to close with saying that the PL can be completely stated without the term "pumping length" (see e.g. here on wikipedia) and if it causes you any trouble, you can always just disregard it.