What is the importance of the condition “| xy | < p” in pumping lemma? [duplicate]

Question

Let L be a language. w $\in$ L , and w could be broken in xyz.

Then if L is regular, there exists a pumping length p such that:

1. |y| $\gt$ 0
2. |xy| $\le$ p
3. $\forall$ i $\ge$ 0, xy$^i$z $\in$ L

I understand that this lemma is based on pigeon hole principle and proof by contradiction.

For conditon (1), I understand that the string we are pumping should be $\ge$ 1.

But I don't understand the significance of condition (2). Why should the legth of xy be equal to or smaller than p . Will not it work otherwise?

Answer 1

$p$ is the number of states in the automaton. As you say, the pumping lemma is about the pigeonhole principle. Suppose that $q$ is the first state that's repeated when you read input $w$. Then $x$ is the string that you read before you reach $q$ for the first time, and $y$ is the string you read between the first and second visits to $q$. Since there are only $p$ states including the start state, you must repeat a state after doing at most $p$ transitions, i.e., reading a string of length at most $p$. So the condition $|xy|\leq p$ comes from the proof.

In reality, when an automaton reads a particular string $w$, there might be many times when it returns to a state that has already been visited. If you take $w\in L$ any split $w=xyz$ with $|y|\geq 1$, such that the automaton is in the same state at the beginning and end of $y$ then $xy^iz$ will be in $L$ for all $i$, regardless of how long $xy$ is.