pumping lemmas

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Two theorems in formal language theory that express necessary conditions for languages to be regular or context-free:

If language L is regular, there exists an integer n such that,

for any word z in L, |z|>n,

there exist u,v,w with z = uvw, v nonempty, |vw|≤n, such that: uvkw ∈ L, for all k≥0 If language L is context-free, there exist integers p and q such that,

z = uvw, v nonempty, |vw|≤n,

uvkw ∈ L, for all k≥0

for any z in L, with |z|>p,

there exist u,v,w,x,y with z = uvwxy, v and x nonempty, |vwx|≤q, such that: uvkwxky ∈ L, for all k ≥ 0 The conditions are used in constructing algorithms for decision problems about regular and context-free grammars, and in proving certain languages are not regular or are not context-free.

z = uvwxy, v and x nonempty,


uvkwxky ∈ L, for all k ≥ 0

Subjects: Computing.

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