Pumping Lemma for CFL

Let $L$ be a CFL. Then $\exists n \in \mathbb{N}$ such that $\forall u \in L$ satisfying $|u| \ge n$, there are strings $v, w, x, y$ and $z$ satisfying:

• $u = vwxyz$
• $|wy| \gt 0$
• $|wxy| \le n$
• For any $m \ge 0$, $vw^mxy^mz$ is in $L$

Ogden’s Lemma

A generalization of the pumping lemma for CFL. Can designate “distinguished” positions of $u$. Can guarantee pumped up portions include at least some of these distinguished positions. Sometimes more convenient to use; can also use when the pumping lemma fails.