# Pumping Lemma for Regular Languages. Q: Why do we care about the Pumping Lemma`; A: We use it to prove that a language is NOT regular.

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 words in a regular language may be pumped —that is, have a middle section of the word repeated an arbitrary number of times—to produce a new word that also lies within the same language.

Pumping Theorem. Let ‘L’ be a regular language. There exist an integer P≥1 such that every string ‘w’ in ‘L’ of length at least ‘P’. It can be written as w = xyz satisfying the following conditions Theorem (Pumping lemma for regular languages) For every regular language L there is a constant k such that every word w 2L of length at least k can be written in the form w = xyz such that the words x, y, and z have the following properties (i) y 6= , (ii) jxyj k, (iii) xyiz 2L for all i 0. Pumping lemma for regular languages Proof. View Pumping lemma.pptx from COMPUTER 888 at Karachi Institute of Economics & Technology, Karachi (City Campus).

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You say “This string x has length p or greater, and does not have a pumpable part near the beginning.” TOC: Pumping Lemma (For Regular Languages)This lecture discusses the concept of Pumping Lemma which is used to prove that a Language is not Regular.Contribut Notes on Pumping Lemma Finite Automata Theory and Formal Languages { TMV027/DIT321 Ana Bove, March 5th 2018 In the course we see two di erent versions of the Pumping lemmas, one for regular languages and one for context-free languages. In what follows we explain how to use these lemmas. 1 Pumping Lemma for Regular Languages Our main goal is to prove a fact about all infinite regular languages that will be helpful in proving that specific languages are nonregular. In particular, this pumping lemma will be the main method we use to prove specific languages are not regular.

Definition Non-Regular Languages and The Pumping Lemma Non-Regular Languages!

## Pumping Lemma is to be applied to show that certain languages are not regular. It should never be used to show a language is regular. If L is regular, it satisfies Pumping Lemma. If L does not satisfy Pumping Lemma, it is non-regular. Method to prove that a language L is not regular. At first, we have to assume that L is regular. So, the pumping lemma should hold for L.

If a regular expression can be constructed to exactly generate the strings in a language, then the language is regular. 4. By Pumping Lemma, there are strings u,v,w such that (i)-(iv) hold. Pick a particular number k ∈ N and argue that uvkw ∈ L, thus yielding our desired contradiction.

### Non-Regular Languages and The Pumping Lemma Non-Regular Languages! • Not every language is a regular language. • However, there are some rules that say "if these languages are regular, so is this one derived from them" •There is also a powerful technique -- the pumping lemma-- that helps us prove a language not to be regular.

• Not every language is a regular language. • However, there are some rules that say "if these languages are regular, so is this one derived from them" •There is also a powerful technique -- the pumping lemma-- that helps us prove a language not to be regular.

Then L has the following property.

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In particular, this pumping lemma will be the main method we use to prove specific languages are not regular. Proof of Pumping Lemma therefore, an FSA cannot be constructed for it. Pumping Lemma states a deep property that all regular languages share. By showing that a language does not have the property stated by the Pumping Lemma, we are guaranteed that it is not regular. 2.

Q: Why do we care about the Pumping Lemma`; A: We use it to prove that a language is NOT regular. If a language is regular, all sufficiently long string in the language can be pumped .

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### There will be no substring that can be pumped in this fashion. Pumping Lemma proof applied to a specific example language. Consider the infinite regular

Let be a CFL.
Using The Pumping Lemma In-Class Examples: Using the Pumping Lemma to show a language L is not regular 5 steps for a proof by contradiction: 1. Assume L is regular. Then, L satisfies the P. Lemma.

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### Recall the pumping lemma for regular languages: Theorem: If Lis a regular language, then there exists a pumping length p where, if sel with Is| Zp, then there exists strings x, y, z such that s = xyz and (i) xy z el for each izo, (ii) ly 21, and (iii) |xy| sp. 4n 2n Prove that A = { c b aln 20 } is not a regular language.

In what follows we explain how to use these lemmas. 1 Pumping Lemma for Regular Languages Our main goal is to prove a fact about all infinite regular languages that will be helpful in proving that specific languages are nonregular. In particular, this pumping lemma will be the main method we use to prove specific languages are not regular. Proof of Pumping Lemma therefore, an FSA cannot be constructed for it. Pumping Lemma states a deep property that all regular languages share.