Satisfying it is a necessary but not sufficient condition for a language to
Satisfying it is a necessary but not sufficient condition for a language to be regular, and its proof is based on the acceptance of a regular language by a deterministic, finite automaton. Another part of the proof uses the Pigeon Hole Principle to show that some state must be revisited. Hence any input of length 1  greater than or equal to n can be subdivided as xyz with length of xy less than or equal to n, length of y greater than or equal to 1, and quantity xy raised to the k times z in regular language L for all natural numbers k. Slightly more complicated when stated for context free languages, FTP, name this famous lemma used to push strings along.
Satisfying it is a necessary but not sufficient condition for a language to be regular, and its proof is based on the acceptance of a regular language by a deterministic, finite automaton.
Pumping lemma