Pumping lemma for context-free languages

In computer science, in particular in formal language theory, the pumping lemma for context-free languages, also known as the Bar-Hillel lemma, is a lemma that gives a property shared by all context-free languages and generalizes the pumping lemma for regular languages. As the pumping lemma does not suffice to guarantee that a language is context-free there are more stringent necessary conditions, such as Ogden's lemma.

If a language L is context-free, then there exists some integer p ≥ 1 (called a "pumping length") such that every string s in L that has a length of p or more symbols (i.e. with |s| ≥ p) can be written as

with substrings u, v, w, x and y, such that

Below is a formal expression of the Pumping Lemma.

${\displaystyle {\begin{array}{l}(\forall L\subseteq \Sigma ^{*})\\\quad ({\mbox{context free}}(L)\Rightarrow \\\quad ((\exists p\geq 1)((\forall s\in L)((|s|\geq p)\Rightarrow \\\quad ((\exists u,v,w,x,y\in \Sigma ^{*})(s=uvwxy\land |vwx|\leq p\land |vx|\geq 1\land (\forall n\geq 0)(uv^{n}wx^{n}y\in L)))))))\end{array}}}$

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