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In the theory of formal languages, the pumping lemma for regular languages is a lemma that describes an essential property of all regular languages. Informally, it says that all sufficiently long strings in a regular language may be pumped—that is, have a middle section of the string repeated an arbitrary number of times—to produce a new string that is also part of the language.
Specifically, the pumping lemma says that for any regular language there exists a constant such that any string in with length at least can be split into three substrings , and (, with being non-empty), such that the strings constructed by repeating zero or more times are still in . This process of repetition is known as "pumping". Moreover, the pumping lemma guarantees that the length of will be at most , imposing a limit on the ways in which may be split.
Languages with a finite number of strings vacuously satisfy the pumping lemma by having equal to the maximum string length in plus one. By doing so, zero strings in have length greater than .
The pumping lemma is useful for disproving the regularity of a specific language in question. It was first proven by Michael Rabin and Dana Scott in 1959, and rediscovered shortly after by Yehoshua Bar-Hillel, Micha A. Perles, and Eli Shamir in 1961, as a simplification of their pumping lemma for context-free languages.