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. The pumping lemma can be used to construct a proof by contradiction that a specific language is not context-free. Conversely, the pumping lemma does not suffice to guarantee that a language is context-free; there are other necessary conditions, such as Ogden's lemma, or the Interchange lemma. If a language is context-free, then there exists some integer (called a "pumping length") such that every string in that has a length of or more symbols (i.e. with ) can be written as with substrings and , such that

1. ,
2. , and
3. for all . Below is a formal expression of the Pumping Lemma. The pumping lemma for context-free languages (called just "the pumping lemma" for the rest of this article) describes a property that all context-free languages are guaranteed to have. The property is a property of all strings in the language that are of length at least , where is a constant—called the pumping length—that varies between context-free languages. Say is a string of length at least that is in the language. The pumping lemma states that can be split into five substrings, , where is non-empty and the length of is at most , such that repeating and the same number of times () in produces a string that is still in the language. It is often useful to repeat zero times, which removes and from the string. This process of "pumping up" with additional copies of and is what gives the pumping lemma its name. Finite languages (which are regular and hence context-free) obey the pumping lemma trivially by having equal to the maximum string length in plus one. As there are no strings of this length the pumping lemma is not violated. The pumping lemma is often used to prove that a given language L is non-context-free, by showing that arbitrarily long strings s are in L that cannot be "pumped" without producing strings outside L.