Pumping lemma for regular languages

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 ${\displaystyle L}$ there exists a constant ${\displaystyle p}$ such that any string ${\displaystyle w}$ in ${\displaystyle L}$ with length at least ${\displaystyle p}$ can be split into three substrings ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ (${\displaystyle w=xyz}$, with ${\displaystyle y}$ being non-empty), such that the strings ${\displaystyle xz,xyz,xyyz,xyyyz,...}$ constructed by repeating ${\displaystyle y}$ zero or more times are still in ${\displaystyle L}$. This process of repetition is known as "pumping". Moreover, the pumping lemma guarantees that the length of ${\displaystyle xy}$ will be at most ${\displaystyle p}$, imposing a limit on the ways in which ${\displaystyle w}$ may be split.

Languages with a finite number of strings vacuously satisfy the pumping lemma by having ${\displaystyle p}$ equal to the maximum string length in ${\displaystyle L}$ plus one. By doing so, zero strings in ${\displaystyle L}$ have length greater than ${\displaystyle p}$.

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.